Day 5 examples
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The document discusses key concepts related to determining whether a function is increasing or decreasing based on its derivative. It defines increasing functions as those where f'(x) is positive over an interval, and decreasing functions as those where f'(x) is negative. It also mentions local maxima and minima points. The document contains a graph of an unlabeled function and asks to (a) list points where the derivative is positive, and (b) list all critical points.
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Day 1a examples
The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
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The document provides instructions for graphing functions with reciprocals. It outlines six steps: 1) Find and sketch any vertical asymptotes where the denominator is zero. 2) Find and sketch any horizontal asymptotes based on the degrees of the numerator and denominator. 3) Find and plot the y-intercept by evaluating f(0). 4) Find the x-intercept by solving the numerator. 5) Use sign analysis to determine where the function is positive and negative. 6) Use smooth curves to complete the graph.
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The document describes how to sketch the graph of y = [f(x)]n, where n is an integer greater than 1. It notes that the graph can be drawn by first sketching y = f(x) and observing that: stationary points and x-intercepts remain the same; [f(x)]n is greater than f(x) if f(x) > 1, and less if f(x) < 1; if n is even, [f(x)]n is always positive; if n is odd, [f(x)]n has the same sign as f(x).
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The document discusses key concepts related to functions including: if the derivative is positive the function is increasing, if negative then decreasing, critical numbers are where the derivative is 0 or doesn't exist, points of inflection are where a curve changes concavity, and an example problem is given to find the local minimum of a cubic function by taking its derivative.
Exercise #13 notes ~ graphing
This document provides instructions for graphing trigonometric transformations in 3 steps: 1) Determine the a, b, c, and d values from the function's factored form. 2) Draw the median position and amplitude. 3) Determine the period and mark points to graph the wave-like function. Examples graph y=3sin(2x)-1, f(x)=sin(1/2x+1), and f(x)=2cos(3x)-2.
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The document discusses transformations of quadratic functions in vertex form f(x) = a(x-h)2 + k. It explains how changing the coefficients a, h, and k affects the graph of the quadratic function. Specifically, it states that changing a widens or narrows the graph, changing h shifts the graph left or right, and changing k shifts the graph up or down. It also provides examples of writing equations for quadratic functions based on given graphs and finding the vertex of a quadratic function in standard form.
Exercise #9 notes
1) Reflections of functions f(x) over the x-axis or y-axis produce new equations by changing the sign of the y-values or x-values respectively.
2) The inverse of a function f(x) is denoted f^-1(x) and is found by switching the x and y values when graphed, effectively reflecting over the line y=x.
3) For example, the inverse of f(x)=2x+5 is found by graphing f^-1(x) which treats x as the output and y as the input.
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1) The document discusses the generalization of derivatives from single-variable calculus to multivariable calculus. In multivariable calculus, a function f(x,y) is differentiable at a point (a,b) if the graph near that point is indistinguishable from a plane.
2) The derivative of a function of two variables f(x,y) at a point (a,b) is represented by the Jacobian matrix Jf(a,b), which can be written as the gradient vector grad f(a,b).
3) The directional derivative of f(x,y) at (a,b) in the direction of a unit vector u is
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The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
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The document provides instructions for graphing functions with reciprocals. It outlines six steps: 1) Find and sketch any vertical asymptotes where the denominator is zero. 2) Find and sketch any horizontal asymptotes based on the degrees of the numerator and denominator. 3) Find and plot the y-intercept by evaluating f(0). 4) Find the x-intercept by solving the numerator. 5) Use sign analysis to determine where the function is positive and negative. 6) Use smooth curves to complete the graph.
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The document describes how to sketch the graph of y = [f(x)]n, where n is an integer greater than 1. It notes that the graph can be drawn by first sketching y = f(x) and observing that: stationary points and x-intercepts remain the same; [f(x)]n is greater than f(x) if f(x) > 1, and less if f(x) < 1; if n is even, [f(x)]n is always positive; if n is odd, [f(x)]n has the same sign as f(x).
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The document discusses key concepts related to functions including: if the derivative is positive the function is increasing, if negative then decreasing, critical numbers are where the derivative is 0 or doesn't exist, points of inflection are where a curve changes concavity, and an example problem is given to find the local minimum of a cubic function by taking its derivative.
Exercise #13 notes ~ graphing
This document provides instructions for graphing trigonometric transformations in 3 steps: 1) Determine the a, b, c, and d values from the function's factored form. 2) Draw the median position and amplitude. 3) Determine the period and mark points to graph the wave-like function. Examples graph y=3sin(2x)-1, f(x)=sin(1/2x+1), and f(x)=2cos(3x)-2.
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The document discusses transformations of quadratic functions in vertex form f(x) = a(x-h)2 + k. It explains how changing the coefficients a, h, and k affects the graph of the quadratic function. Specifically, it states that changing a widens or narrows the graph, changing h shifts the graph left or right, and changing k shifts the graph up or down. It also provides examples of writing equations for quadratic functions based on given graphs and finding the vertex of a quadratic function in standard form.
Exercise #9 notes
1) Reflections of functions f(x) over the x-axis or y-axis produce new equations by changing the sign of the y-values or x-values respectively.
2) The inverse of a function f(x) is denoted f^-1(x) and is found by switching the x and y values when graphed, effectively reflecting over the line y=x.
3) For example, the inverse of f(x)=2x+5 is found by graphing f^-1(x) which treats x as the output and y as the input.
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1) The document discusses the generalization of derivatives from single-variable calculus to multivariable calculus. In multivariable calculus, a function f(x,y) is differentiable at a point (a,b) if the graph near that point is indistinguishable from a plane.
2) The derivative of a function of two variables f(x,y) at a point (a,b) is represented by the Jacobian matrix Jf(a,b), which can be written as the gradient vector grad f(a,b).
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This document provides examples for graphing reciprocals of functions. It explains that for reciprocals, smaller numbers on the x-axis become larger on the y-axis, and vice versa. An example graphs the reciprocal of f(x)=1/x^2. It notes that reciprocals can be written in the form f(x)=k/x+h, where k and h are determined from the original function. Further examples graph the reciprocals of f(x)=2/x+4 and f(x)=sin(x).
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- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
Integration. area undera curve
The document discusses rules of integration, including:
1) The definite integral calculates the area under a curve between two bounds a and b.
2) If the function is negative in some intervals, the integral is the sum of the areas of regions where the function is positive minus the areas where it is negative.
3) The fundamental theorem of calculus relates the definite integral to antiderivatives.
Day 1 examples 2
This document discusses horizontal shifts of functions. It explains that a term like (x - h) shifts a function f(x) right by h units, while (x + h) shifts it left by h units. It provides examples of sketching graphs after horizontal shifts and finding domains and ranges of shifted functions. The document also covers describing shifted functions in words and finding equations of translated absolute value graphs.
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docx
The document contains a 30-question math exam covering topics like functions, graphs, equations, inequalities, logarithms, and other math concepts. It includes multiple choice, short answer, and show work questions assessing skills like domain and range, solving equations, graphing, composites, inverses, lines, maximizing profit, and more. Students must demonstrate mathematical reasoning and problem-solving abilities.
AP Calculus Slides December 10, 2007
This document contains several calculus practice problems related to applications of derivatives. It includes questions about determining where a function is increasing or decreasing based on its graph, finding values of derivatives from a table, determining concavity from a table of derivatives, approximating derivatives, identifying conditions for local extrema, sketching graphs with horizontal asymptotes, and determining domains and ranges.
Advanced Functions Unit 1
Here are the key steps to find the instantaneous rate of change using a graphing calculator:
1. Graph the function over the appropriate domain.
2. Use the arrow keys to move the cursor to the point where you want to find the instantaneous rate of change.
3. Press the TRACE button and select the tangent option.
4. The calculator will display the slope of the tangent line, which is the instantaneous rate of change at that point.
5. For example, if finding the IROC at x=1 for the function f(x) = x3, you would:
a) Graph f(x) = x3
b) Use arrows to move cursor
Second Derivative Information
The second derivative of a function f(x) provides information about the concavity and points of inflection of f(x). It indicates the intervals where f(x) is concave up or down. A curve is concave up where the second derivative is positive and concave down where the second derivative is negative. Points of inflection are where the concavity changes from up to down or vice versa, and occur where the second derivative is zero or undefined.
Day 6 examples
The document discusses identifying critical points, inflection points, and points of concavity for a function based on its graph over the domain [0,10]. Critical points are where the derivative is zero or undefined. Inflection points occur where the concavity changes from upward to downward or vice versa. To find inflection points, locate where the second derivative is zero or undefined and check that the second derivative has opposite signs on either side.
Intro to Matlab programming
MATLAB is a programming tool that simplifies programming compared to languages like C and C#. The document introduces basic MATLAB functions like help, inputting matrices and vectors, matrix operations, loops, conditional statements, and graphs. It also covers symbolic math, Laplace transforms, Fourier transforms, and other domains and transforms. The overall document serves as an introduction to essential MATLAB programming concepts and capabilities.
237654933 mathematics-t-form-6
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Core2.beamer
The document discusses different types of transformations of graphs: translations, reflections, and one-way stretches. It provides examples and explanations of how translations move a graph horizontally or vertically, reflections flip the graph across an axis, and stretches change the scale of the graph in the x or y direction. Key transformations include: y = f(x) + b for vertical translation; y = f(x - a) for horizontal translation; y = -f(x) for reflection across the x-axis; and y = df(x) for a stretch by a scale factor d in the y direction.
Calculus- Basics
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
4.3 The Definite Integral
The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.
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MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docx
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This document contains 5 problems involving finding terms in expansions of polynomials. The problems involve finding specific terms that contain a given power of x in expansions of polynomials such as (3x^4 - 1)^9, (-x^3 + 2)^6, (x + 1)^3x, (x + 1)^x, and determining the value of m if one term in the expansion of (2x - m)^7 is -262500x^2y^5.
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The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.
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This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
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Permutations refer to arrangements of objects in a definite order. Some key points:
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Day 2 examples u7w14
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
Day 1 examples u7w14
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Day 4 examples u6w14
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Day 3 examples u6w14
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
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The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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Day 5 examples
- 2. Critical Numbers Increasing = graph moves up as we move to the right Decreasing = graph moves down as we move to the right If f '(x) > 0 on an interval (a, b) then f is increasing on that interval. If f '(x) < 0 on an interval (a, b) then f is decreasing on that interval.
- 4. local/relative maximum points local/relative minimum point
- 6. ) increasing? ) increasing fastest? ) decreasing? ) decreasing fastest? Note: -2 ≤ x ≤ 3 up? down? Where does local max? min?
- 10. The graph of a function ƒ with domain [0, 10] is shown in the figure. Several points are labeled. (a) List all labeled points at which ƒ'(x) > 0. (b) List all critical points of ƒ.