The document discusses slopes and derivatives. It defines slope as the ratio of the "rise" over the "run" between two points on a line. For a curve, the slope at a point is defined as the slope of the tangent line at that point. The derivative at a point is also called the slope of the tangent line and represents the instantaneous rate of change of the function at that point. The document provides an example of using slopes to calculate rates like velocity and fuel efficiency from distance and time measurements.
The document discusses limits and derivatives. It explains that in calculating the derivative of f(x)=x^2 - 2x + 2, the slope formula was simplified. As h approaches 0, the chords slide towards the tangent line, so the slope at (x,f(x)) is 2x-2. It then provides definitions and explanations for what it means for a variable to approach 0 from the right, left, or in general, to clarify the procedure of obtaining slopes using limits.
The document discusses the concept of slope and the difference quotient formula for calculating slope. It defines a function f(x) and points P(x,f(x)) and Q(x+h, f(x+h)) on the graph of f(x). The slope of the cord connecting points P and Q is given by the difference quotient (f(x+h) - f(x))/h. An example problem calculates this slope for the specific points P(2,2) and Q(2.2,2.44) on the parabola y=x^2 - 2x + 2.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses properties of derivatives and how they relate to limits. It states that the sum, difference, and constant multiple rules for limits directly apply to differentiation. However, the product and quotient rules for limits do not directly apply to differentiation, which has more complicated product and quotient rules. Elementary functions are defined in terms of a few basic formulas and operations. The document then examines the sum and constant multiple rules for derivatives in more detail, proving them using limits. It also provides a geometric illustration of how the derivative of a sum is equal to the sum of the derivatives.
1) The document discusses derivatives as rates of change, using the example of a stone thrown straight up.
2) It is found that the stone will stay in the air for 6 seconds, reaching its maximum height of 144 feet after 3 seconds.
3) The derivative of the height function D(t) represents the instantaneous rate of change of height, or speed, at each time t. This rate varies throughout the stone's trajectory.
The document discusses exponential and logarithmic functions. Exponential functions of the form f(x) = b^x are called exponential functions in base b. Logarithmic functions log_b(y) represent the exponent x needed to raise the base b to a power to get the output y. The exponential form b^x = y and logarithmic form x = log_b(y) describe the same relationship between the base b, exponent x, and output y. Questions can be translated between these forms by rewriting the exponential expression as a logarithm or vice versa. Examples demonstrate rewriting expressions and graphing logarithmic functions.
- The derivative of a function f(x) represents the instantaneous rate of change of the output y with respect to the input x. It is equivalent to the slope of the tangent line and the amount of change in y for a 1 unit change in x.
- For a linear price-demand function of y = f(x) chickens sold given price x, the derivative of the revenue function R(x) = x*f(x) represents how revenue changes with a 1 unit change in price.
- The price that maximizes revenue occurs when the derivative of the revenue function R'(x) is 0, as this is where revenue is no longer increasing or decreasing with small changes in price.
The document discusses limits of fractional expressions as the variable approaches certain values. It provides four basic facts about the limits of fractions of elementary functions: (1) if the numerator and denominator have defined limits, the fractional limit is the fraction of the limits; (2) if the numerator is bounded and the denominator diverges, the fractional limit is 0; (3) if the numerator diverges and the denominator is bounded, the fractional limit is infinity; (4) if both the numerator and denominator have limits of 0 or infinity, the fractional limit is inconclusive. It emphasizes that an undefined fractional limit does not necessarily mean the limit is inconclusive - it may simply not exist. Rationalizing expressions can sometimes resolve inconclusive fractional limits
The document discusses limits and derivatives. It explains that in calculating the derivative of f(x)=x^2 - 2x + 2, the slope formula was simplified. As h approaches 0, the chords slide towards the tangent line, so the slope at (x,f(x)) is 2x-2. It then provides definitions and explanations for what it means for a variable to approach 0 from the right, left, or in general, to clarify the procedure of obtaining slopes using limits.
The document discusses the concept of slope and the difference quotient formula for calculating slope. It defines a function f(x) and points P(x,f(x)) and Q(x+h, f(x+h)) on the graph of f(x). The slope of the cord connecting points P and Q is given by the difference quotient (f(x+h) - f(x))/h. An example problem calculates this slope for the specific points P(2,2) and Q(2.2,2.44) on the parabola y=x^2 - 2x + 2.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses properties of derivatives and how they relate to limits. It states that the sum, difference, and constant multiple rules for limits directly apply to differentiation. However, the product and quotient rules for limits do not directly apply to differentiation, which has more complicated product and quotient rules. Elementary functions are defined in terms of a few basic formulas and operations. The document then examines the sum and constant multiple rules for derivatives in more detail, proving them using limits. It also provides a geometric illustration of how the derivative of a sum is equal to the sum of the derivatives.
1) The document discusses derivatives as rates of change, using the example of a stone thrown straight up.
2) It is found that the stone will stay in the air for 6 seconds, reaching its maximum height of 144 feet after 3 seconds.
3) The derivative of the height function D(t) represents the instantaneous rate of change of height, or speed, at each time t. This rate varies throughout the stone's trajectory.
The document discusses exponential and logarithmic functions. Exponential functions of the form f(x) = b^x are called exponential functions in base b. Logarithmic functions log_b(y) represent the exponent x needed to raise the base b to a power to get the output y. The exponential form b^x = y and logarithmic form x = log_b(y) describe the same relationship between the base b, exponent x, and output y. Questions can be translated between these forms by rewriting the exponential expression as a logarithm or vice versa. Examples demonstrate rewriting expressions and graphing logarithmic functions.
- The derivative of a function f(x) represents the instantaneous rate of change of the output y with respect to the input x. It is equivalent to the slope of the tangent line and the amount of change in y for a 1 unit change in x.
- For a linear price-demand function of y = f(x) chickens sold given price x, the derivative of the revenue function R(x) = x*f(x) represents how revenue changes with a 1 unit change in price.
- The price that maximizes revenue occurs when the derivative of the revenue function R'(x) is 0, as this is where revenue is no longer increasing or decreasing with small changes in price.
The document discusses limits of fractional expressions as the variable approaches certain values. It provides four basic facts about the limits of fractions of elementary functions: (1) if the numerator and denominator have defined limits, the fractional limit is the fraction of the limits; (2) if the numerator is bounded and the denominator diverges, the fractional limit is 0; (3) if the numerator diverges and the denominator is bounded, the fractional limit is infinity; (4) if both the numerator and denominator have limits of 0 or infinity, the fractional limit is inconclusive. It emphasizes that an undefined fractional limit does not necessarily mean the limit is inconclusive - it may simply not exist. Rationalizing expressions can sometimes resolve inconclusive fractional limits
The document summarizes different types of derivatives. It discusses simple derivatives where there is one input and output, and defines them. It then discusses implicit derivatives where a relationship between two variables is given and the derivative of one with respect to the other is sought using implicit differentiation. An example finds the derivative of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. Reciprocal relationships between the derivatives are noted.
3.2 implicit equations and implicit differentiationmath265
The document discusses implicit equations and implicit differentiation. It begins by explaining the difference between explicit and implicit forms of equations, using the example of y=1/x which can be written explicitly as y=1/x or implicitly as xy=1. It then introduces the concept of implicit differentiation, which involves taking the derivative of an implicit equation with respect to x and solving for the derivative of y with respect to x (y’). This allows one to find the slope of the curve at a point, even if the explicit form of the relation between x and y is difficult to determine from the implicit equation.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
This document discusses rules for computing derivatives of functions. It begins by listing existing derivative rules and defining notation. It then derives and presents rules for the derivatives of trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant. An example problem demonstrates finding the derivative of the tangent function using previous rules.
There are two systems for measuring angles: the degree system and the radian system. The degree system divides a full circle into 360 equal angles of 1 degree each. The radian system defines an angle as the arc length cut out by the angle on a unit circle of radius 1, where a full circle corresponds to 2π radians. While the degree system is commonly used, the radian system is preferred in mathematics due to its relationship to circle geometry formulas involving arc lengths and wedge areas.
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
6 comparison statements, inequalities and intervals ymath260
The document discusses how to translate comparison statements and phrases into mathematical inequalities. It explains that real numbers can be represented on a number line, with positive numbers to the right of zero and negative numbers to the left. Common comparisons like "greater than", "less than", "at least", and "at most" are then defined in terms of inequalities. For example, "x is greater than a" is written as "a < x", and "x is at most b" is written as "x ≤ b". Compound comparisons are also addressed, such as "x is more than a but no more than b" being written as "a < x ≤ b".
1.2 review on algebra 2-sign charts and inequalitiesmath265
The document discusses sign charts and inequalities. It explains that sign charts can be used to determine if expressions are positive or negative by factoring them and evaluating at given values of x. Examples are provided to demonstrate how to construct a sign chart by: 1) solving for where the expression equals 0, 2) marking these values on a number line, and 3) evaluating the expression at sample points in each segment to determine the signs in between values where the expression equals 0. The sign chart then indicates the ranges where the expression is positive, negative or zero.
14 graphs of factorable rational functions xmath260
The document discusses graphs of rational functions. It defines rational functions as functions of the form R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials. It describes how vertical asymptotes occur where the denominator Q(x) is zero. The graph runs along either side of vertical asymptotes, going up or down depending on the sign chart. There are four cases for how the graph behaves at a vertical asymptote. The document uses examples to illustrate graphing rational functions and determining vertical asymptotes. It also mentions horizontal asymptotes will be discussed.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document defines the derivative of a function f(x) as the limit of the difference quotient (f(x+h) - f(x))/h as h approaches 0. This represents the slope of the tangent line to the function f(x) at the point x. An example is worked out where the derivative of the function f(x) = x^2 - 2x + 2 is calculated to be 2x - 2. The derivative is denoted by f'(x) and represents the instantaneous rate of change of the function at the point x.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, where the dividend is divided by the divisor to obtain a quotient and remainder. Synthetic division is simpler but can only be used to divide a polynomial by a monomial. The key points are then demonstrated through worked examples of long division.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions involving variables and operations. Polynomial expressions are algebraic expressions that can be written in the form anxn + an-1xn-1 + ... + a1x + a0, where the ai coefficients are numbers. The document gives examples of factoring polynomials using formulas like a3b3 = (ab)(a2ab + b2). Factoring polynomials makes it easier to calculate outputs and simplify expressions for operations like addition and subtraction.
This document discusses first degree functions and linear equations. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential/log formulas. Linear equations of the form Ax + By = C represent straight lines that can be graphed by finding the x- and y-intercepts. If an equation contains only one variable, it represents a vertical or horizontal line. The slope-intercept form y = mx + b is introduced, where m is the slope and b is the y-intercept. Slope is defined as the ratio of the rise over the run between two points on a line.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
This document discusses optimization problems in real-world applications and the role of derivatives. It provides examples of functions that may or may not have extrema over an interval. The extrema theorem for continuous functions states that a continuous function over a closed interval will have both an absolute maximum and minimum. Extrema can occur where the derivative is zero, where the derivative is undefined, or at the endpoints. Examples are provided to illustrate the different types of extrema.
The document discusses Cavalieri's principle, which states that two regions with identical cross-sectional areas have the same volume. It demonstrates this principle by showing how the volume of a circle can be determined by unwinding its concentric rings into a triangle of equal area. The 3D version of the principle is also described, indicating that solids with identical cross-sectional areas have the same volume. Examples of prisms and cones are provided.
The document discusses continuity of functions and graphs. It defines a continuous function as one where the graph is unbroken within its domain. A function is discontinuous if its graph is broken. Continuity at a point x=a can be determined by comparing the left and right limits of the function at a to the actual value of the function at a. If the limits are equal to the function value, it is continuous from that side. The document provides examples of functions that are right continuous, left continuous, or discontinuous at various points to illustrate these concepts.
The document discusses various chain rules for derivatives, including:
- The power chain rule: [up]' = pup−1(u)'
- Trigonometric chain rules: [sin(u)]' = cos(u)(u)', [cos(u)]' = −sin(u)(u)'
- Examples are provided to demonstrate applying the chain rules to find derivatives of more complex functions like y = sin(x3) and y = sin3(x). Repeated application of the appropriate chain rule at each step is often required.
The document summarizes different types of derivatives. It discusses simple derivatives where there is one input and output, and defines them. It then discusses implicit derivatives where a relationship between two variables is given and the derivative of one with respect to the other is sought using implicit differentiation. An example finds the derivative of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. Reciprocal relationships between the derivatives are noted.
3.2 implicit equations and implicit differentiationmath265
The document discusses implicit equations and implicit differentiation. It begins by explaining the difference between explicit and implicit forms of equations, using the example of y=1/x which can be written explicitly as y=1/x or implicitly as xy=1. It then introduces the concept of implicit differentiation, which involves taking the derivative of an implicit equation with respect to x and solving for the derivative of y with respect to x (y’). This allows one to find the slope of the curve at a point, even if the explicit form of the relation between x and y is difficult to determine from the implicit equation.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
This document discusses rules for computing derivatives of functions. It begins by listing existing derivative rules and defining notation. It then derives and presents rules for the derivatives of trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant. An example problem demonstrates finding the derivative of the tangent function using previous rules.
There are two systems for measuring angles: the degree system and the radian system. The degree system divides a full circle into 360 equal angles of 1 degree each. The radian system defines an angle as the arc length cut out by the angle on a unit circle of radius 1, where a full circle corresponds to 2π radians. While the degree system is commonly used, the radian system is preferred in mathematics due to its relationship to circle geometry formulas involving arc lengths and wedge areas.
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
6 comparison statements, inequalities and intervals ymath260
The document discusses how to translate comparison statements and phrases into mathematical inequalities. It explains that real numbers can be represented on a number line, with positive numbers to the right of zero and negative numbers to the left. Common comparisons like "greater than", "less than", "at least", and "at most" are then defined in terms of inequalities. For example, "x is greater than a" is written as "a < x", and "x is at most b" is written as "x ≤ b". Compound comparisons are also addressed, such as "x is more than a but no more than b" being written as "a < x ≤ b".
1.2 review on algebra 2-sign charts and inequalitiesmath265
The document discusses sign charts and inequalities. It explains that sign charts can be used to determine if expressions are positive or negative by factoring them and evaluating at given values of x. Examples are provided to demonstrate how to construct a sign chart by: 1) solving for where the expression equals 0, 2) marking these values on a number line, and 3) evaluating the expression at sample points in each segment to determine the signs in between values where the expression equals 0. The sign chart then indicates the ranges where the expression is positive, negative or zero.
14 graphs of factorable rational functions xmath260
The document discusses graphs of rational functions. It defines rational functions as functions of the form R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials. It describes how vertical asymptotes occur where the denominator Q(x) is zero. The graph runs along either side of vertical asymptotes, going up or down depending on the sign chart. There are four cases for how the graph behaves at a vertical asymptote. The document uses examples to illustrate graphing rational functions and determining vertical asymptotes. It also mentions horizontal asymptotes will be discussed.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document defines the derivative of a function f(x) as the limit of the difference quotient (f(x+h) - f(x))/h as h approaches 0. This represents the slope of the tangent line to the function f(x) at the point x. An example is worked out where the derivative of the function f(x) = x^2 - 2x + 2 is calculated to be 2x - 2. The derivative is denoted by f'(x) and represents the instantaneous rate of change of the function at the point x.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, where the dividend is divided by the divisor to obtain a quotient and remainder. Synthetic division is simpler but can only be used to divide a polynomial by a monomial. The key points are then demonstrated through worked examples of long division.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions involving variables and operations. Polynomial expressions are algebraic expressions that can be written in the form anxn + an-1xn-1 + ... + a1x + a0, where the ai coefficients are numbers. The document gives examples of factoring polynomials using formulas like a3b3 = (ab)(a2ab + b2). Factoring polynomials makes it easier to calculate outputs and simplify expressions for operations like addition and subtraction.
This document discusses first degree functions and linear equations. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential/log formulas. Linear equations of the form Ax + By = C represent straight lines that can be graphed by finding the x- and y-intercepts. If an equation contains only one variable, it represents a vertical or horizontal line. The slope-intercept form y = mx + b is introduced, where m is the slope and b is the y-intercept. Slope is defined as the ratio of the rise over the run between two points on a line.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
This document discusses optimization problems in real-world applications and the role of derivatives. It provides examples of functions that may or may not have extrema over an interval. The extrema theorem for continuous functions states that a continuous function over a closed interval will have both an absolute maximum and minimum. Extrema can occur where the derivative is zero, where the derivative is undefined, or at the endpoints. Examples are provided to illustrate the different types of extrema.
The document discusses Cavalieri's principle, which states that two regions with identical cross-sectional areas have the same volume. It demonstrates this principle by showing how the volume of a circle can be determined by unwinding its concentric rings into a triangle of equal area. The 3D version of the principle is also described, indicating that solids with identical cross-sectional areas have the same volume. Examples of prisms and cones are provided.
The document discusses continuity of functions and graphs. It defines a continuous function as one where the graph is unbroken within its domain. A function is discontinuous if its graph is broken. Continuity at a point x=a can be determined by comparing the left and right limits of the function at a to the actual value of the function at a. If the limits are equal to the function value, it is continuous from that side. The document provides examples of functions that are right continuous, left continuous, or discontinuous at various points to illustrate these concepts.
The document discusses various chain rules for derivatives, including:
- The power chain rule: [up]' = pup−1(u)'
- Trigonometric chain rules: [sin(u)]' = cos(u)(u)', [cos(u)]' = −sin(u)(u)'
- Examples are provided to demonstrate applying the chain rules to find derivatives of more complex functions like y = sin(x3) and y = sin3(x). Repeated application of the appropriate chain rule at each step is often required.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
The document discusses properties of logarithms. It begins by recalling rules of exponents and their corresponding rules of logarithms. Four basic logarithm rules are presented: 1) logb(1) = 0, 2) logb(xy) = logb(x) + logb(y), 3) logb(x/y) = logb(x) - logb(y), 4) logb(xt) = tlogb(x). It then works through an example problem to demonstrate using these rules to write the logarithm of a expression in terms of logarithms of its variables. It concludes by noting that logarithms and exponentials are inverse functions, so logb(bx) =
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....loniyakrishn
The document discusses directional derivatives and gradients. It defines a directional derivative as the instantaneous rate of change of a multivariate function moving in a given direction. It also defines the gradient as a vector whose components are the partial derivatives of the function, and whose direction points in the direction of greatest increase of the function. The gradient allows one to calculate directional derivatives using a dot product relationship. Examples are provided to illustrate directional derivatives, gradients, and their applications in problems involving slopes and rates of change.
IVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdf42Rnu
Unit-1 covers topics related to error analysis, graphing, and logarithms. It discusses types of errors, propagation of errors through addition, subtraction, multiplication, division, and powers. It also defines standard deviation and provides examples of calculating it. Graphing concepts like dependent and independent variables, linear and nonlinear functions, and plotting graphs from equations are explained. Logarithm rules and properties are also introduced.
11. amplitude, phase shift and period of trig formulas-xmath260
The document discusses amplitude, period, and phase shift of waves. It defines amplitude as the distance from the waistline of a wave to its peak or trough. Period is the length of time or space for a wave to complete one cycle. Vertically stretching or compressing a sine or cosine graph changes its amplitude. Horizontally compressing or stretching changes its period. For example, y=sin(2x) has half the period of y=sin(x) because as x varies from 0 to π, 2x varies from 0 to 2π, compressing the wave horizontally.
This document discusses differentials and how they relate to differentiable functions. Some key points:
1. The differential of an independent variable x is defined as dx, which is equal to the increment Δx. The differential of a dependent variable y is defined as dy = f'(x) dx, where f'(x) is the derivative of the function.
2. Differentials allow approximations of changes in a function using derivatives, such as estimating errors or finding approximate roots.
3. Rules are provided for finding differentials of common functions using differentiation formulas. Examples demonstrate using differentials to estimate changes and approximate values.
The document provides an overview of Calculus I taught by Professor Matthew Leingang at New York University. It outlines key topics that will be covered in the course, including different classes of functions, transformations of functions, and compositions of functions. The first assignments are due on January 31 and February 2, with first recitations on February 3. The document uses examples to illustrate concepts like linear functions, other polynomial functions, and trigonometric functions. It also explains how vertical and horizontal shifts can transform the graph of a function.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
1) Derivatives relate the rates of change of position, velocity, and acceleration. Velocity is the derivative of position and measures rate of change of displacement. Acceleration is the derivative of velocity and measures the rate of change of velocity.
2) The Mean Value Theorem states that for a continuous function over an interval, there exists at least one point where the slope of the tangent line equals the slope of the secant line between the endpoints.
3) A function's derivatives provide information about the behavior of the original function. The first derivative relates to slope and critical points where the function is increasing/decreasing. The second derivative indicates points of inflection where the concavity changes.
Vectors have both magnitude and direction, while scalars only have magnitude. There are two main methods for adding vectors graphically: the head-to-tail method and the parallelogram method. Vectors can also be represented and added using their horizontal and vertical components. The dot product of two vectors yields a scalar that indicates the cosine of the angle between the vectors, while the vector product yields a vector that is perpendicular to both original vectors and whose magnitude depends on the angle between them.
1. The document discusses the concept of derivatives and how to calculate them. It defines key terms like increment, average rate of change, and instantaneous rate of change.
2. Methods are provided for calculating the derivative of various types of functions, including polynomials, rational functions, and functions with roots. Examples are worked through step-by-step.
3. The relationship between derivatives and tangent lines is explained, with derivatives defined as the instantaneous rate of change or slope of the curve at a point.
Basic algebra, trig and calculus needed for physics.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
TIU CET Review Math Session 6 - part 2 of 2youngeinstein
1. The document provides a review of math concepts for a college entrance exam, including functions, trigonometric functions, exponential and logarithmic functions.
2. It reviews concepts like evaluating functions, adding and composing functions, finding roots and intercepts of functions, and properties of trigonometric, exponential and logarithmic functions.
3. The document provides examples and problems to solve related to these various math concepts as a study guide for the exam.
This document provides information about graphing linear equations. It begins by defining a linear equation as one whose solutions fall on a straight line. It explains how to identify if an equation is linear based on whether a constant change in the x-value corresponds to a constant change in the y-value. The document then gives examples of graphing equations and determining if they are linear based on whether their graphs form a straight line. It also discusses using tables to list the x and y-values of points that satisfy the equation.
The document discusses various mathematical concepts related to functions and graphs including:
1) Transformations of graphs such as translations, reflections, and rotations. It also discusses parent functions and their derivatives.
2) Examples of graphing functions after applying transformations to translate, scale, or reflect the original graphs. Equations are provided for the transformed graphs.
3) Theorems related to how statistics of data change after translations or scale changes. For example, the mean, median and mode change proportionally but variance, standard deviation, and range change in specific ways.
4) Concepts involving inverse functions, including using the horizontal line test to determine if an inverse is a function and notations for inverse functions
The document defines slope as the ratio of the rise (change in y-values) to the run (change in x-values) between two points on a line. It provides the exact formula for calculating slope as the change in y-values divided by the change in x-values. Examples are given to demonstrate calculating the slopes of various lines, with positive slopes for lines passing through Quadrants I and III and negative slopes for lines passing through Quadrants II and IV.
The document provides information about calculating the slope of a line from a graph or two points, including examples and practice problems. Key terms are defined, such as rise, run, slope, dependent and independent variables. An example problem demonstrates how to find the slope from a table of gas costs and gallons and interpret what the slope represents. A lesson quiz provides practice finding slopes and interpreting what they represent based on graphs and tables.
The document defines slope as the ratio of the "rise" over the "run" between two points on a line. Specifically, if the points are (x1, y1) and (x2, y2), then the slope m is equal to (y2 - y1) / (x2 - x1). It also discusses how to calculate the slope of a line given two points, and how the slope indicates whether a line rises or falls from left to right. Lines between the first and third quadrants have positive slopes, while lines between the second and fourth quadrants have negative slopes.
1. The document discusses vector calculus concepts including the gradient, divergence, curl, and theorems relating integrals.
2. It defines the curl of a vector field A as the maximum circulation of A per unit area and provides expressions for curl in Cartesian, cylindrical and spherical coordinates.
3. Stokes's theorem is described as relating a line integral around a closed path to a surface integral of the curl over the enclosed surface, allowing transformation between different integral types.
The document discusses acceleration and related concepts:
- Acceleration is the change in velocity per unit of time and is a vector quantity. It results from an applied force and is proportional to the force's magnitude.
- Velocity is speed in a given direction, while speed is the distance traveled per time and does not consider direction.
- Average acceleration is calculated as the change in velocity divided by the time interval. Instantaneous acceleration is the slope of the velocity-time graph at an instant.
- Examples demonstrate calculating average speed and acceleration from initial and final velocities and time intervals. Direction and signs of displacement, velocity, and acceleration must be considered carefully.
characteristic of function, average rate chnage, instant rate chnage.pptxPallaviGupta66118
A power function is a polynomial function of the form y=xn, where n is a real number constant called the exponent. Power functions can be odd or even degree. Odd degree power functions have point symmetry about the origin, while even degree functions may have line symmetry. The instantaneous rate of change at a point is the slope of the tangent line to the curve at that point and can be estimated from graphs, tables of values, or equations.
The document discusses limits and how they are used to calculate the derivative of a function. It defines what it means for a sequence to approach a limit from the right or left side. Graphs and examples are provided to illustrate these concepts. The key rules for calculating limits are outlined, such as using algebra to split limits into their constituent parts. Common types of obvious limits are also stated, such as limits of constants or products involving constants.
The document discusses the concept of limits and clarifies the notation used to describe sequences approaching a number. It explains that saying "x approaches 0 from the right side" means the sequence values only become smaller than 0 after a finite number of terms. Similarly, approaching from the left means only finitely many terms are greater than 0. The direction a sequence approaches a number affects limits like the limit of |x|/x as x approaches 0.
This document discusses two sections, Section 3.1 and Section 3.3, but provides no details about the content or topics covered in either section. The document gives the section numbers and titles but no other informative or descriptive text.
The document describes how to calculate the volume of a solid object using Cavalieri's principle. It involves partitioning the solid into thin cross-sectional slices and approximating the volume of each slice as a cylinder with the slice's cross-sectional area and thickness. The total volume is then approximated as the sum of the cylindrical slice volumes. As the number of slices approaches infinity, this sum approaches the actual volume calculated as the integral of the cross-sectional area function over the solid's distance range.
The document discusses calculating the area of a region R. It introduces using a ruler x to measure the span of R from x=a to x=b. It defines the cross-sectional length L(x) and partitions the interval [a,b] into subintervals. The Riemann sum of the areas of approximating rectangles is shown to approach the actual area of R, defined as the definite integral of L(x) from a to b. As an example, it calculates the area between the curves y=-x^2+2x and y=x^2 by finding the interval spans from 0 to 1 and taking the integral of the difference of the functions.
The document discusses the substitution method of integration. It explains that while the derivative of an elementary function is another elementary function, the antiderivative may not be. There are two main integration methods: substitution and integration by parts. Substitution reverses the chain rule by letting u be a function of x with derivative u', then substituting u for x and replacing dx with du/u' in the integral.
The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.
4.5 continuous functions and differentiable functionsmath265
The document discusses continuous and differentiable functions. It defines elementary functions as those constructed using basic operations like addition and multiplication. Continuous functions over a closed interval are bounded and have absolute maximum and minimum values. The Intermediate Value Theorem states that a continuous function takes on all values between its minimum and maximum. Differentiable functions are continuous. Rolle's Theorem says that if a differentiable function is equal at the endpoints of an interval, its derivative is zero somewhere in between.
The document discusses related rates problems. It begins by using resizing a rectangle on a computer screen as an example to demonstrate how the rates of change of the length (L) and width (W) relate to the rate of change of the area (A). The key steps are: (1) the area A is given by A=LW, (2) take the derivative of both sides, (3) use the product rule and chain rule to obtain A'=L'W+LW', (4) plug in the given rates of L' and W' to solve for A'.
The document then provides examples to demonstrate how to set up and solve related rates problems by translating the given rates into derivatives, applying
The document summarizes different types of derivatives:
Simple derivatives involve a single input and output. Implicit derivatives are taken for equations with two or more variables, treating one as the independent variable. An example finds derivatives of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. The derivatives are related by the reciprocal relationship in differential notation.
This document contains 20 math word problems involving rates of change of quantities like distance, area, radius, and volume over time. The problems involve concepts like expanding derivatives, rectangles changing size, cars moving at intersections, distances between moving objects, water filling and draining from tanks, ladders on houses, waves expanding in water, balloons deflating, and water filling triangular troughs. Rates of change are calculated for variables like length, width, area, distance, radius, and volume at specific values over time.
The document contains 10 multi-part exercises involving calculating rates of change, finding maximums and optima, and approximating changes in functions. The exercises involve concepts like linear price-demand functions, surface area and volume relationships for geometric objects, and force functions related to physics concepts like gravity and electric force.
1. The document provides instructions for using calculus concepts like derivatives and integrals to approximate values. It contains 14 problems involving finding derivatives, using derivatives to approximate values, finding volumes with integrals, and using Newton's method to find roots of functions.
2. The final problem asks to use Newton's method in Excel to find the two roots of the function y = ex - 2x - 2 that exist between -3 and 3 to 5 decimal places, and then justify that the approximations are correct.
This document contains 16 multi-part math problems involving optimization of functions, geometry, and physics. The problems cover topics like finding extrema of functions, finding points on lines, maximizing areas of geometric shapes given constraints, minimizing materials needed to construct cylinders and fences, and finding positions of maximum or minimum values of physical quantities like force and illumination.
This document discusses two applications of tangent lines: differentials and linear approximation, and finding the tangent line T(b) at a nearby point b. It explains that the tangent line T(x) at point (a, f(a)) is given by T(x) = f'(a)(x - a) + f(a). The slope f'(a) is identified with the derivative dy/dx. There are two ways to find T(b): directly using T(x), or by finding the differential ΔT = dy and using ΔT + f(a) = T(b).
The document discusses how derivatives can represent rates of change. It states that given a function f(x), the derivative f'(a) is equivalent to the slope of the tangent line at x=a, the instantaneous rate of change of y with respect to x at x=a, and the amount of change in y for a 1 unit change in x at x=a. It then provides an example using a price-demand function for chickens, finding that the maximum revenue of $1152 occurs at a price of $10 per chicken.
1. Graph and analyze the critical points, extrema, inflection points, intervals of increasing/decreasing, and intervals of concave up/down for 10 functions.
2. Review homework on finding derivatives using the definition of the difference quotient and evaluating limits. Find the derivatives of 6 functions.
3. Use implicit differentiation to find the derivative of one function defined implicitly and to find points with tangent lines of slope 1 for another implicit function.
4. Find the second derivatives of two functions.
1) The document provides a tutorial on using formulas in Excel, including how to enter formulas, use relative and absolute cell references, perform calculations on ranges of cells, and sum columns of data.
2) It includes steps to enter sample data, calculate values like x-squared and frequencies multiplied by x and x-squared, and use formulas to automatically calculate those values down a column.
3) The tutorial concludes with instructions to sum the sample data columns, enter the student's name, save the Excel file, and provide a printout.
The document discusses using the second derivative to identify extrema and classify flat points on a graph of y=f(x). It defines terms for the second derivative, explaining that if f''(x)>0, the slope f'(x) is increasing, meaning a downhill point is getting less steep and an uphill point is getting more steep. For a maximum point M, the curve must flatten out with f'(x) approaching 0+ and f'(x) becoming increasingly negative after M, resulting in f''(M)<0.
The document discusses derivatives and graphs. It defines interval notation used to indicate whether points are included or excluded from intervals. It then explains that the derivative of a function f(x) at a point x, f'(x), represents the slope of the tangent line to the graph of f(x) at (x, f(x)). Finally, it notes that points where the derivative is 0 are called critical points, as the tangent line is flat at these points.
AI-Powered Food Delivery Transforming App Development in Saudi Arabia.pdfTechgropse Pvt.Ltd.
In this blog post, we'll delve into the intersection of AI and app development in Saudi Arabia, focusing on the food delivery sector. We'll explore how AI is revolutionizing the way Saudi consumers order food, how restaurants manage their operations, and how delivery partners navigate the bustling streets of cities like Riyadh, Jeddah, and Dammam. Through real-world case studies, we'll showcase how leading Saudi food delivery apps are leveraging AI to redefine convenience, personalization, and efficiency.
Building Production Ready Search Pipelines with Spark and MilvusZilliz
Spark is the widely used ETL tool for processing, indexing and ingesting data to serving stack for search. Milvus is the production-ready open-source vector database. In this talk we will show how to use Spark to process unstructured data to extract vector representations, and push the vectors to Milvus vector database for search serving.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Your One-Stop Shop for Python Success: Top 10 US Python Development Providersakankshawande
Simplify your search for a reliable Python development partner! This list presents the top 10 trusted US providers offering comprehensive Python development services, ensuring your project's success from conception to completion.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
AI 101: An Introduction to the Basics and Impact of Artificial IntelligenceIndexBug
Imagine a world where machines not only perform tasks but also learn, adapt, and make decisions. This is the promise of Artificial Intelligence (AI), a technology that's not just enhancing our lives but revolutionizing entire industries.
Have you ever been confused by the myriad of choices offered by AWS for hosting a website or an API?
Lambda, Elastic Beanstalk, Lightsail, Amplify, S3 (and more!) can each host websites + APIs. But which one should we choose?
Which one is cheapest? Which one is fastest? Which one will scale to meet our needs?
Join me in this session as we dive into each AWS hosting service to determine which one is best for your scenario and explain why!
Programming Foundation Models with DSPy - Meetup SlidesZilliz
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
Ocean lotus Threat actors project by John Sitima 2024 (1).pptxSitimaJohn
Ocean Lotus cyber threat actors represent a sophisticated, persistent, and politically motivated group that poses a significant risk to organizations and individuals in the Southeast Asian region. Their continuous evolution and adaptability underscore the need for robust cybersecurity measures and international cooperation to identify and mitigate the threats posed by such advanced persistent threat groups.
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
How does your privacy program stack up against your peers? What challenges are privacy teams tackling and prioritizing in 2024?
In the fifth annual Global Privacy Benchmarks Survey, we asked over 1,800 global privacy professionals and business executives to share their perspectives on the current state of privacy inside and outside of their organizations. This year’s report focused on emerging areas of importance for privacy and compliance professionals, including considerations and implications of Artificial Intelligence (AI) technologies, building brand trust, and different approaches for achieving higher privacy competence scores.
See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
This webinar will review:
- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
Fueling AI with Great Data with Airbyte WebinarZilliz
This talk will focus on how to collect data from a variety of sources, leveraging this data for RAG and other GenAI use cases, and finally charting your course to productionalization.
Monitoring and Managing Anomaly Detection on OpenShift.pdfTosin Akinosho
Monitoring and Managing Anomaly Detection on OpenShift
Overview
Dive into the world of anomaly detection on edge devices with our comprehensive hands-on tutorial. This SlideShare presentation will guide you through the entire process, from data collection and model training to edge deployment and real-time monitoring. Perfect for those looking to implement robust anomaly detection systems on resource-constrained IoT/edge devices.
Key Topics Covered
1. Introduction to Anomaly Detection
- Understand the fundamentals of anomaly detection and its importance in identifying unusual behavior or failures in systems.
2. Understanding Edge (IoT)
- Learn about edge computing and IoT, and how they enable real-time data processing and decision-making at the source.
3. What is ArgoCD?
- Discover ArgoCD, a declarative, GitOps continuous delivery tool for Kubernetes, and its role in deploying applications on edge devices.
4. Deployment Using ArgoCD for Edge Devices
- Step-by-step guide on deploying anomaly detection models on edge devices using ArgoCD.
5. Introduction to Apache Kafka and S3
- Explore Apache Kafka for real-time data streaming and Amazon S3 for scalable storage solutions.
6. Viewing Kafka Messages in the Data Lake
- Learn how to view and analyze Kafka messages stored in a data lake for better insights.
7. What is Prometheus?
- Get to know Prometheus, an open-source monitoring and alerting toolkit, and its application in monitoring edge devices.
8. Monitoring Application Metrics with Prometheus
- Detailed instructions on setting up Prometheus to monitor the performance and health of your anomaly detection system.
9. What is Camel K?
- Introduction to Camel K, a lightweight integration framework built on Apache Camel, designed for Kubernetes.
10. Configuring Camel K Integrations for Data Pipelines
- Learn how to configure Camel K for seamless data pipeline integrations in your anomaly detection workflow.
11. What is a Jupyter Notebook?
- Overview of Jupyter Notebooks, an open-source web application for creating and sharing documents with live code, equations, visualizations, and narrative text.
12. Jupyter Notebooks with Code Examples
- Hands-on examples and code snippets in Jupyter Notebooks to help you implement and test anomaly detection models.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
3. (x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1
= the difference in the
heights of the points.
Δx = x2 – x1
= the difference in the
run of the points.
Δy
Δx=The slope m is the ratio of the “rise” to the “run”.
Slopes and Derivatives
4. (x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1
= the difference in the
heights of the points.
Δx = x2 – x1
= the difference in the
run of the points.
Δy
Δx=The slope m is the ratio of the “rise” to the “run”.
The slope measures the tilt of a line in relation to the
horizon, that is, the steepness in relation to the x
axis.
Slopes and Derivatives
5. (x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1
= the difference in the
heights of the points.
Δx = x2 – x1
= the difference in the
run of the points.
Δy
Δx=The slope m is the ratio of the “rise” to the “run”.
The slope measures the tilt of a line in relation to the
horizon, that is, the steepness in relation to the x
axis. Therefore horizontal lines have its steepness or
slope = 0 .
Slopes and Derivatives
6. (x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1
= the difference in the
heights of the points.
Δx = x2 – x1
= the difference in the
run of the points.
Δy
Δx=The slope m is the ratio of the “rise” to the “run”.
* http://www.mathwarehouse.com/algebra/linear_equation/interactive-
slope.php
The slope measures the tilt of a line in relation to the
horizon, that is, the steepness in relation to the x
axis. Therefore horizontal lines have its steepness or
slope = 0 . Steeper lines have “larger” slopes*.
Slopes and Derivatives
8. Algebra of Slope
Δy = y2 – y1
= the difference in the outputs y
Slopes and Derivatives
9. Algebra of Slope
Δy = y2 – y1
= the difference in the outputs y
Δx = x2 – x1
= the difference in the inputs x
Slopes and Derivatives
10. Algebra of Slope
Δy = y2 – y1
= the difference in the outputs y
Δx = x2 – x1
= the difference in the inputs x
Algebraically the slope
m = Δy/Δx = Δy : Δx is the ratio
the difference in the outputs the difference in the inputs:
Slopes and Derivatives
11. Algebra of Slope
Δy = y2 – y1
= the difference in the outputs y
Δx = x2 – x1
= the difference in the inputs x
Algebraically the slope
m = Δy/Δx = Δy : Δx is the ratio
the difference in the outputs the difference in the inputs:
The units of this ratio are (units of y) / (units of x).
Slopes and Derivatives
12. Algebra of Slope
Δy = y2 – y1
= the difference in the outputs y
Δx = x2 – x1
= the difference in the inputs x
Algebraically the slope
m = Δy/Δx = Δy : Δx is the ratio
the difference in the outputs the difference in the inputs:
The units of this ratio are (units of y) / (units of x).
This is also the amount of change in y for each unit
change in x.
Slopes and Derivatives
13. Algebra of Slope
Δy = y2 – y1
= the difference in the outputs y
Δx = x2 – x1
= the difference in the inputs x
Algebraically the slope
m = Δy/Δx = Δy : Δx is the ratio
the difference in the outputs the difference in the inputs:
The units of this ratio are (units of y) / (units of x).
This is also the amount of change in y for each unit
change in x. The ratio “2 eggs : 3 cakes” is the same
as “2/3 egg per cake”.
Slopes and Derivatives
14. Algebra of Slope
Δy = y2 – y1
= the difference in the outputs y
Δx = x2 – x1
= the difference in the inputs x
Algebraically the slope
m = Δy/Δx = Δy : Δx is the ratio
the difference in the outputs the difference in the inputs:
The units of this ratio are (units of y) / (units of x).
This is also the amount of change in y for each unit
change in x. The ratio “2 eggs : 3 cakes” is the same
as “2/3 egg per cake”. The slope m is “the amount of
change of y if x changes by one unit.
Slopes and Derivatives
15. Example A. We had 6 gallons of gas at the start of a
trip and the odometer was registered at 75,000 miles.
Two hours later, the gas gauge indicated there were
3 gallons left and the odometer was at 75,300 miles.
Let x = the gas–tank indicator reading
y = the odometer reading
t = the time in hours. Find the following slopes.
Slopes and Derivatives
16. Example A. We had 6 gallons of gas at the start of a
trip and the odometer was registered at 75,000 miles.
Two hours later, the gas gauge indicated there were
3 gallons left and the odometer was at 75,300 miles.
Let x = the gas–tank indicator reading
y = the odometer reading
t = the time in hours. Find the following slopes.
a. Compare the measurements of the fuel amount x
versus the distance y traveled, or (x, y),
Slopes and Derivatives
17. Example A. We had 6 gallons of gas at the start of a
trip and the odometer was registered at 75,000 miles.
Two hours later, the gas gauge indicated there were
3 gallons left and the odometer was at 75,300 miles.
Let x = the gas–tank indicator reading
y = the odometer reading
t = the time in hours. Find the following slopes.
a. Compare the measurements of the fuel amount x
versus the distance y traveled, or (x, y). We have
(6, 75000), (3, 75300).
Slopes and Derivatives
18. The slope m = (75,300 – 75,000) / (3 – 6)
= –100 mpg.
Example A. We had 6 gallons of gas at the start of a
trip and the odometer was registered at 75,000 miles.
Two hours later, the gas gauge indicated there were
3 gallons left and the odometer was at 75,300 miles.
Let x = the gas–tank indicator reading
y = the odometer reading
t = the time in hours. Find the following slopes.
a. Compare the measurements of the fuel amount x
versus the distance y traveled, or (x, y). We have
(6, 75000), (3, 75300).
Slopes and Derivatives
19. The slope m = (75,300 – 75,000) / (3 – 6)
= –100 mpg.
So the distance–to–fuel rate or the fuel efficiency
is 100 miles per gallon.
Example A. We had 6 gallons of gas at the start of a
trip and the odometer was registered at 75,000 miles.
Two hours later, the gas gauge indicated there were
3 gallons left and the odometer was at 75,300 miles.
Let x = the gas–tank indicator reading
y = the odometer reading
t = the time in hours. Find the following slopes.
a. Compare the measurements of the fuel amount x
versus the distance y traveled, or (x, y). We have
(6, 75000), (3, 75300).
Slopes and Derivatives
20. b. Compare the measurements of the time t versus
the distance y traveled, or (t, y).
Slopes and Derivatives
21. b. Compare the measurements of the time t versus
the distance y traveled, or (t, y). we have
(0, 75000), (2, 75300).
Slopes and Derivatives
22. The slope n = (75,300 – 70,000) / (2 – 0)
= 150 mph.
b. Compare the measurements of the time t versus
the distance y traveled, or (t, y). we have
(0, 75000), (2, 75300).
Slopes and Derivatives
23. So the distance–to–time rate (or the velocity) per hour
is 150 miles per hour.
The slope n = (75,300 – 70,000) / (2 – 0)
= 150 mph.
b. Compare the measurements of the time t versus
the distance y traveled, or (t, y). we have
(0, 75000), (2, 75300).
Slopes and Derivatives
24. So the distance–to–time rate (or the velocity) per hour
is 150 miles per hour.
The slope n = (75,300 – 70,000) / (2 – 0)
= 150 mph.
b. Compare the measurements of the time t versus
the distance y traveled, or (t, y). we have
(0, 75000), (2, 75300).
c. Compare the measurements of the time t versus
the amount of fuel x, or (t, x),
Slopes and Derivatives
25. So the distance–to–time rate (or the velocity) per hour
is 150 miles per hour.
The slope n = (75,300 – 70,000) / (2 – 0)
= 150 mph.
b. Compare the measurements of the time t versus
the distance y traveled, or (t, y). we have
(0, 75000), (2, 75300).
the slope n = (6 – 3) / (0 – 2)
= –1.5 gph.
c. Compare the measurements of the time t versus
the amount of fuel x, or (t, x). We have
(0, 6), (2, 3). The rate of change is
Slopes and Derivatives
26. So the distance–to–time rate (or the velocity) per hour
is 150 miles per hour.
The slope n = (75,300 – 70,000) / (2 – 0)
= 150 mph.
b. Compare the measurements of the time t versus
the distance y traveled, or (t, y). we have
(0, 75000), (2, 75300).
So the fuel–to–time rate of the fuel consumption per
hour is 1.50 gallon per hour.
c. Compare the measurements of the time t versus
the amount of fuel x, or (t, x). We have
(0, 6), (2, 3). The rate of change is
Slopes and Derivatives
the slope n = (6 – 3) / (0 – 2)
= –1.5 gph.
27. Question: What are the reciprocals of the above
rates and what do they measure?
Slopes and Derivatives
28. Question: What are the reciprocals of the above
rates and what do they measure?
Slopes measure steepness of straight lines.
Slopes and Derivatives
29. Question: What are the reciprocals of the above
rates and what do they measure?
Slopes measure steepness of straight lines.
We want to extend this system of geometric
measurement to measuring “steepness” of curves.
Slopes and Derivatives
30. Question: What are the reciprocals of the above
rates and what do they measure?
Slopes measure steepness of straight lines.
We want to extend this system of geometric
measurement to measuring “steepness” of curves.
A curve has “ups” and “downs”
so a curve has different
“slopes” at different points.
Slopes and Derivatives
31. Question: What are the reciprocals of the above
rates and what do they measure?
x
P
y= f(x)
Slopes measure steepness of straight lines.
We want to extend this system of geometric
measurement to measuring “steepness” of curves.
Q
A curve has “ups” and “downs”
so a curve has different
“slopes” at different points.
This may be seen in the figure
shown at points P and Q.
Slopes and Derivatives
32. Question: What are the reciprocals of the above
rates and what do they measure?
x
P
y= f(x)
Slopes measure steepness of straight lines.
We want to extend this system of geometric
measurement to measuring “steepness” of curves.
Q
A curve has “ups” and “downs”
so a curve has different
“slopes” at different points.
This may be seen in the figure
shown at points P and Q.
Obviously the “slope” at the point P should be positive,
and the “slope” at the point Q should be negative.
Slopes and Derivatives
33. Question: What are the reciprocals of the above
rates and what do they measure?
x
P
y= f(x)
Slopes measure steepness of straight lines.
We want to extend this system of geometric
measurement to measuring “steepness” of curves.
Q
A curve has “ups” and “downs”
so a curve has different
“slopes” at different points.
This may be seen in the figure
shown at points P and Q.
We define the “slope at a point P on the curve y = f(x)”
to be “the slope of the tangent line to y = f(x) at P” .
Obviously the “slope” at the point P should be positive,
and the “slope” at the point Q should be negative.
Slopes and Derivatives
35. Slopes and Derivatives
Derivatives
If f(x) is an elementary function,
then the “slope” of the tangent is
well defined for “most” of the
points on its graph y = f(x).
36. Slopes and Derivatives
x
P
y= f(x)
Derivatives
If f(x) is an elementary function,
then the “slope” of the tangent is
well defined for “most” of the
points on its graph y = f(x).
37. Slopes and Derivatives
x
P
y= f(x)
Derivatives
If f(x) is an elementary function,
then the “slope” of the tangent is
well defined for “most” of the
points on its graph y = f(x).
The slope at a point P is also
called the derivative at P.
38. Slopes and Derivatives
x
P
y= f(x)
Derivatives
If f(x) is an elementary function,
then the “slope” of the tangent is
well defined for “most” of the
points on its graph y = f(x).
The slope at a point P is also
called the derivative at P.
By “well defined” we mean that the geometric notion of
“the tangent line at P” is intuitive and unambiguous so
its slope is unambiguous.
39. Slopes and Derivatives
x
P
y= f(x)
Derivatives
By “well defined” we mean that the geometric notion of
“the tangent line at P” is intuitive and unambiguous so
its slope is unambiguous.
We will examine the notion of the “tangent
line” in the next section. For now, we
accept the tangent line intuitively as a
straight line that leans against y = f(x) at P
as shown.
If f(x) is an elementary function,
then the “slope” of the tangent is
well defined for “most” of the
points on its graph y = f(x).
The slope at a point P is also
called the derivative at P.
40. Slopes and Derivatives
x
P
y= f(x)
Derivatives
By “well defined” we mean that the geometric notion of
“the tangent line at P” is intuitive and unambiguous so
its slope is unambiguous.
Furthermore, it is well defined because we are able to
compute the slopes of tangents at different locations
algebraically.
If f(x) is an elementary function,
then the “slope” of the tangent is
well defined for “most” of the
points on its graph y = f(x).
The slope at a point P is also
called the derivative at P.
41. Slopes and Derivatives
x
P
y= f(x)
Derivatives
By “well defined” we mean that the geometric notion of
“the tangent line at P” is intuitive and unambiguous so
its slope is unambiguous.
Furthermore, it is well defined because we are able to
compute the slopes of tangents at different locations
algebraically. We will carry out these computations for
the 2nd degree example from the last section.
If f(x) is an elementary function,
then the “slope” of the tangent is
well defined for “most” of the
points on its graph y = f(x).
The slope at a point P is also
called the derivative at P.
42. Example B.
Given f(x) = x2 – 2x + 2
a. Find the slope of the cord connecting the points
(x, f(x)) and (x+h,f(x+h)) with x = 2 and h = 0.2.
Slopes and Derivatives
43. Example B.
Given f(x) = x2 – 2x + 2
a. Find the slope of the cord connecting the points
(x, f(x)) and (x+h,f(x+h)) with x = 2 and h = 0.2.
Substitute the values of x and h,
we are find the slopes of the cord
connecting (2, f(2)=2) and
(2.2, f(2.2)). (2.2, f(2.2))
(2, 2)
2 2.2
0.2
Slopes and Derivatives
44. Example B.
Given f(x) = x2 – 2x + 2
a. Find the slope of the cord connecting the points
(x, f(x)) and (x+h,f(x+h)) with x = 2 and h = 0.2.
f(x+h) – f(x)
h
Substitute the values of x and h,
we are find the slopes of the cord
connecting (2, f(2)=2) and
(2.2, f(2.2)). Its slope is
f(2.2) – f(2)
0.2
= (2, 2)
2 2.2
0.2
(2.2, f(2.2))
Slopes and Derivatives
45. Example B.
Given f(x) = x2 – 2x + 2
a. Find the slope of the cord connecting the points
(x, f(x)) and (x+h,f(x+h)) with x = 2 and h = 0.2.
f(x+h) – f(x)
h
Substitute the values of x and h,
we are find the slopes of the cord
connecting (2, f(2)=2) and
(2.2, f(2.2)). Its slope is
f(2.2) – f(2)
0.2
=
=
2.44 – 2
0.2
= 2.2
(2, 2)
2 2.2=
0.44
0.2
0.44
0.2
slope m = 2.2
(2.2, f(2.2))
Slopes and Derivatives
46. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (2, 2) and
(2+h, f(2+h)).
Slopes and Derivatives
47. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (2, 2) and
(2+h, f(2+h)).
(2+h, f(2+h)
(2, 2)
2 2 + h
h
f(2+h)–f(2)
Slopes and Derivatives
48. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (2, 2) and
(2+h, f(2+h)).
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2 at x = 2.
(2+h, f(2+h)
(2, 2)
2 2 + h
h
f(2+h)–f(2)
Slopes and Derivatives
49. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (2, 2) and
(2+h, f(2+h)).
f(2+h) – f(2)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2 at x = 2.
(2+h, f(2+h)
(2, 2)
2 2 + h
h
f(2+h)–f(2)
Slopes and Derivatives
50. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (2, 2) and
(2+h, f(2+h)).
f(2+h) – f(2)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2 at x = 2.
=
[(2+h)2 – 2(2+h) + 2] – (2)
h
(2+h, f(2+h)
(2, 2)
2 2 + h
h
f(2+h)–f(2)
Slopes and Derivatives
51. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (2, 2) and
(2+h, f(2+h)).
f(2+h) – f(2)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2 at x = 2.
=
[(2+h)2 – 2(2+h) + 2] – (2)
h
h2 + 2h
h
=
(2+h, f(2+h)
(2, 2)
2 2 + h
h
f(2+h)–f(2)
Slopes and Derivatives
52. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (2, 2) and
(2+h, f(2+h)).
f(2+h) – f(2)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2 at x = 2.
=
[(2+h)2 – 2(2+h) + 2] – (2)
h
h2 + 2h
h
= h + 2
=
(2+h, f(2+h)
(2, 2)
2 2 + h
h
slope = h + 2
f(2+h)–f(2)
Slopes and Derivatives
53. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
Slopes and Derivatives
54. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
(2, 2)
2
y = x2–2x+2
Slopes and Derivatives
55. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
The cord is fixed at the base–point
(2, 2) and the other end depends on
the value h.
(2, 2)
2
y = x2–2x+2
Slopes and Derivatives
56. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
(2+h, f(2+h)
(2, 2)
2 2 + h
h
f(2+h)–f(2)
The cord is fixed at the base–point
(2, 2) and the other end depends on
the value h.
y = x2–2x+2
Slopes and Derivatives
57. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
(2+h, f(2+h)
(2, 2)
2 2 + h
h
f(2+h)–f(2)
The cord is fixed at the base–point
(2, 2) and the other end depends on
the value h.
y = x2–2x+2
Slopes and Derivatives
58. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
(2+h, f(2+h)
(2, 2)
2 2 + h
f(2+h)–f(2)
The cord is fixed at the base–point
(2, 2) and the other end depends on
the value h. As h varies, we obtained
different cords with different slopes.
h
y = x2–2x+2
Slopes and Derivatives
59. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
(2+h, f(2+h)
(2, 2)
2 2 + h
f(2+h)–f(2)
The cord is fixed at the base–point
(2, 2) and the other end depends on
the value h. As h varies, we obtained
different cords with different slopes.
As h gets larger, the cords deviates
away from the tangent line at (2, 2)
and as h gets smaller the
corresponding cords swing and settle
toward the tangent line. h
y = x2–2x+2
Slopes and Derivatives
60. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
(2+h, f(2+h)
(2, 2)
2 2 + h
f(2+h)–f(2)
The cord is fixed at the base–point
(2, 2) and the other end depends on
the value h. As h varies, we obtained
different cords with different slopes.
As h gets larger, the cords deviates
away from the tangent line at (2, 2)
and as h gets smaller the
corresponding cords swing and settle
toward the tangent line. These cords
have slopes 2 + h.
slope = 2 + h
h
y = x2–2x+2
Slopes and Derivatives
61. c. Now we deduce the slope at the point (2, 2) in the
following geometric argument.
(2+h, f(2+h)
(2, 2)
2 2 + h
slope = 2 + h
f(2+h)–f(2)
The cord is fixed at the base–point
(2, 2) and the other end depends on
the value h. As h varies, we obtained
different cords with different slopes.
As h gets larger, the cords deviates
away from the tangent line at (2, 2)
and as h gets smaller the
corresponding cords swing and settle
toward the tangent line. These cords
have slopes 2 + h. Hence as
h “shrinks” to 0, the slope or the
derivative at (2, 2) must be 2.
h
y = x2–2x+2
Slopes and Derivatives
62. d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
Slopes and Derivatives
63. Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)).
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
Slopes and Derivatives
64. (x+h, f(x+h)
(x, f(x))
x x + h
f(x+h)–f(x)
Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)).
h
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
y = x2–2x+2
Slopes and Derivatives
65. (x+h, f(x+h)
(x, f(x))
x x + h
f(x+h)–f(x)
Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)). The difference quotient is
h
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
f(x+h) – f(x)
h
y = x2–2x+2
Slopes and Derivatives
66. (x+h, f(x+h)
(x, f(x))
x x + h
f(x+h)–f(x)
Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)). The difference quotient is
h
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
f(x+h) – f(x)
h
=
(x+h)2 – 2(x+h) + 2 – [x2 – 2x + 2]
h
y = x2–2x+2
Slopes and Derivatives
67. (x+h, f(x+h)
(x, f(x))
x x + h
f(x+h)–f(x)
Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)). The difference quotient is
h
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
f(x+h) – f(x)
h
=
(x+h)2 – 2(x+h) + 2 – [x2 – 2x + 2]
h
2xh – 2h + h2
h
=
y = x2–2x+2
Slopes and Derivatives
68. (x+h, f(x+h)
(x, f(x))
x x + h
f(x+h)–f(x)
Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)). The difference quotient is
h
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
f(x+h) – f(x)
h
=
(x+h)2 – 2(x+h) + 2 – [x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h
=
y = x2–2x+2
slope = 2x–2+h
Slopes and Derivatives
69. (x+h, f(x+h)
(x, f(x))
x x + h
f(x+h)–f(x)
Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)). The difference quotient is
h
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
f(x+h) – f(x)
h
=
(x+h)2 – 2(x+h) + 2 – [x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h
=
As h shrinks to 0, the slopes of the
cords approach the value 2x – 2
y = x2–2x+2
slope = 2x–2+h
Slopes and Derivatives
70. (x+h, f(x+h)
(x, f(x))
x x + h
f(x+h)–f(x)
Connect the cord at the base–point
(x, f(x)) to the other end point at
(x+h, f(x+h)). The difference quotient is
h
d. Simplify the difference quotient of f(x) then find the
slope at the point (x, f(x)) – as a formula in x.
f(x+h) – f(x)
h
=
(x+h)2 – 2(x+h) + 2 – [x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h.
=
As h shrinks to 0, the slopes of the
cords approach the value 2x – 2
which must be the slope at (x, f(x)).
y = x2–2x+2
slope = 2x–2+h
Slopes and Derivatives
72. Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
Slopes and Derivatives
73. Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
y = x2–2x+2
Slopes and Derivatives
74. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
y = x2–2x+2
Slopes and Derivatives
75. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
y = x2–2x+2
Slopes and Derivatives
76. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
y = x2–2x+2
slope at x
= 2x – 2
Slopes and Derivatives
77. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
This slope–formula is the derivative of
f(x) and it is written as f ’(x)
y = x2–2x+2
slope at x
= 2x – 2
Slopes and Derivatives
78. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
This slope–formula is the derivative of
f(x) and it is written as f ’(x) – it’s read
as “f prime of x”.
y = x2–2x+2
slope at x
= 2x – 2
Slopes and Derivatives
79. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
This slope–formula is the derivative of
f(x) and it is written as f ’(x) – it’s read
as “f prime of x”. So the derivative of
f(x) = x2 – 2x + 2 is f ’(x) = 2x – 2.
y = x2–2x+2
slope at x
= 2x – 2
Slopes and Derivatives
80. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
This slope–formula is the derivative of
f(x) and it is written as f ’(x) – it’s read
as “f prime of x”. So the derivative of
f(x) = x2 – 2x + 2 is f ’(x) = 2x – 2.
y = x2–2x+2
The name derivative came from the
fact that f’(x) = 2x – 2 is derived from
f(x) = x2 – 2x + 2.
slope at x
= 2x – 2
Slopes and Derivatives
81. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
This slope–formula is the derivative of
f(x) and it is written as f ’(x) – it’s read
as “f prime of x”. So the derivative of
f(x) = x2 – 2x + 2 is f ’(x) = 2x – 2.
y = x2–2x+2
The name derivative came from the
fact that f’(x) = 2x – 2 is derived from
f(x) = x2 – 2x + 2. The derivative f ’(x) = 2x – 2 tells
us the slopes of f(x) = x2 – 2x + 2.
slope at x
= 2x – 2
Slopes and Derivatives
82. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
This slope–formula is the derivative of
f(x) and it is written as f ’(x) – it’s read
as “f prime of x”. So the derivative of
f(x) = x2 – 2x + 2 is f ’(x) = 2x – 2.
y = x2–2x+2
The name derivative came from the
fact that f’(x) = 2x – 2 is derived from
f(x) = x2 – 2x + 2. The derivative f ’(x) = 2x – 2 tells
us the slopes of f(x) = x2 – 2x + 2.
For example the slope at x = 2 is f ’(2) = 2,
at x = 3 is f ’(3) = 4, etc…
slope at x
= 2x – 2
Slopes and Derivatives
83. (x, f(x))
x
Let’s summarize the result.
If f(x) = x2 – 2x + 2, then the slope at
the point (x, f(x)) is 2x – 2.
This slope–formula is the derivative of
f(x) and it is written as f ’(x) – it’s read
as “f prime of x”. So the derivative of
f(x) = x2 – 2x + 2 is f ’(x) = 2x – 2.
y = x2–2x+2
The name derivative came from the
fact that f’(x) = 2x – 2 is derived from
f(x) = x2 – 2x + 2. The derivative f ’(x) = 2x – 2 tells
us the slopes of f(x) = x2 – 2x + 2.
For example the slope at x = 2 is f ’(2) = 2,
at x = 3 is f ’(3) = 4, etc… The derivative f ’(x) is the
extension of the concept of slopes of lines to curves.
slope at x
= 2x – 2
Slopes and Derivatives
84. The first topic of the calculus course is derivatives.
Slopes and Derivatives
85. The first topic of the calculus course is derivatives.
We will examine derivative algebraically by
* developing the language of derivative which
makes the concept more rigorous
Slopes and Derivatives
86. The first topic of the calculus course is derivatives.
We will examine derivative algebraically by
* developing the language of derivative which
makes the concept more rigorous
* developing the computation techniques for finding
derivatives of all elementary functions
Slopes and Derivatives
87. The first topic of the calculus course is derivatives.
We will examine derivative algebraically by
* developing the language of derivative which
makes the concept more rigorous
* developing the computation techniques for finding
derivatives of all elementary functions
Slopes and Derivatives
We will examine derivative geometrically
88. The first topic of the calculus course is derivatives.
We will examine derivative algebraically by
* developing the language of derivative which
makes the concept more rigorous
* developing the computation techniques for finding
derivatives of all elementary functions
Slopes and Derivatives
We will examine derivative geometrically by
* developing the relations between derivatives f ’(x)
and the shape of the graph of y = f(x)
89. The first topic of the calculus course is derivatives.
We will examine derivative algebraically by
* developing the language of derivative which
makes the concept more rigorous
* developing the computation techniques for finding
derivatives of all elementary functions
Slopes and Derivatives
We will examine derivative geometrically by
* developing the relations between derivatives f ’(x)
and the shape of the graph of y = f(x)
* developing the computation procedures using f ’(x)
to locate special positions on y = f(x)
90. The first topic of the calculus course is derivatives.
We will examine derivative algebraically by
* developing the language of derivative which
makes the concept more rigorous
* developing the computation techniques for finding
derivatives of all elementary functions
Slopes and Derivatives
We will examine derivative geometrically by
* developing the relations between derivatives f ’(x)
and the shape of the graph of y = f(x)
* developing the computation procedures using f ’(x)
to locate special positions on y = f(x)
We will also investigate the applications of the
derivatives which include problems of optimization,
rates of change and numerical methods.