The document discusses approximations of areas under curves using the trapezoidal rule. It introduces the trapezoidal rule formula and shows how it can be used to approximate areas with multiple intervals by summing the areas of individual trapezoids. In general, the area is approximated as the average of the initial and final y-values plus twice the sum of the other y-values, divided by two. An example demonstrates applying the rule to approximate the area under a curve between 0 and 2 using 4 intervals.
The document describes the trapezoidal rule for approximating the area under a curve. The trapezoidal rule works by dividing the area into trapezoid sections and summing their individual areas. In general, the area is approximated as the average of the initial and final y-values plus twice the sum of the internal y-values, divided by the number of sections. An example applies this to estimate the area under a given curve divided into 4 intervals.
The document defines and provides examples of several types of functions including:
1) Constant functions where f(x) = a for all values of x.
2) Linear functions of the form f(x) = ax + b.
3) Quadratic functions of the form f(x) = ax2 + bx + c.
4) Polynomial functions which are the sum of terms with variables raised to various powers.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function g∘f. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
The document discusses methods to calculate the volume of a rectangular pyramid. It provides two methods: (1) using similar triangles and (2) using coordinate geometry. The volume formula derived is: V = (1/3)ab(h-z)^3/h, where a and b are the lengths of the base and h-z is the height.
This document provides solutions to review problems involving combining functions through addition, subtraction, multiplication, division, and composition. Some key examples include:
- Sketching the graphs of f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x) given the graphs of f(x) and g(x)
- Writing explicit equations for combinations of functions and determining their domains and ranges
- Evaluating composite functions like f(g(x)) and g(f(x)) given definitions of f(x) and g(x)
- Determining if two functions are inverses using their compositions
The document lists 4 formulas relevant to a Math 1230 course:
1) Euler's method for numerical integration of differential equations.
2) Formulas for finding the centroid (center of mass) of a plane region and the average value of a function over that region.
3) Taylor series representation of functions, expressing a function as a sum of terms involving its derivatives.
4) Rules for differentiating and integrating power series representations of functions.
The document discusses partial derivative equations and homogeneous functions. It defines:
- The partial derivative of a function f(x,y) with respect to x and y at a point.
- Clairaut's theorem, which relates mixed partial derivatives to commutative properties.
- The Laplace equation in 2D and 3D, which relates second order partial derivatives of a function.
- Conditions for a function of two variables to be homogeneous of a given degree.
The document describes the trapezoidal rule for approximating the area under a curve. The trapezoidal rule works by dividing the area into trapezoid sections and summing their individual areas. In general, the area is approximated as the average of the initial and final y-values plus twice the sum of the internal y-values, divided by the number of sections. An example applies this to estimate the area under a given curve divided into 4 intervals.
The document defines and provides examples of several types of functions including:
1) Constant functions where f(x) = a for all values of x.
2) Linear functions of the form f(x) = ax + b.
3) Quadratic functions of the form f(x) = ax2 + bx + c.
4) Polynomial functions which are the sum of terms with variables raised to various powers.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function g∘f. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
The document discusses methods to calculate the volume of a rectangular pyramid. It provides two methods: (1) using similar triangles and (2) using coordinate geometry. The volume formula derived is: V = (1/3)ab(h-z)^3/h, where a and b are the lengths of the base and h-z is the height.
This document provides solutions to review problems involving combining functions through addition, subtraction, multiplication, division, and composition. Some key examples include:
- Sketching the graphs of f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x) given the graphs of f(x) and g(x)
- Writing explicit equations for combinations of functions and determining their domains and ranges
- Evaluating composite functions like f(g(x)) and g(f(x)) given definitions of f(x) and g(x)
- Determining if two functions are inverses using their compositions
The document lists 4 formulas relevant to a Math 1230 course:
1) Euler's method for numerical integration of differential equations.
2) Formulas for finding the centroid (center of mass) of a plane region and the average value of a function over that region.
3) Taylor series representation of functions, expressing a function as a sum of terms involving its derivatives.
4) Rules for differentiating and integrating power series representations of functions.
The document discusses partial derivative equations and homogeneous functions. It defines:
- The partial derivative of a function f(x,y) with respect to x and y at a point.
- Clairaut's theorem, which relates mixed partial derivatives to commutative properties.
- The Laplace equation in 2D and 3D, which relates second order partial derivatives of a function.
- Conditions for a function of two variables to be homogeneous of a given degree.
1. The document provides solutions to trigonometric and algebraic equations. It solves equations involving sin, cos, tan functions as well as polynomials with variables x, y, z.
2. The algebraic equations section involves solving polynomials for single variables x or y, as well as systems of equations with variables x and y.
3. The trigonometric equations section expresses the general solutions to equations involving sin, cos, and tan functions in terms of radian angles n*pi/b, where b is the denominator and n is any integer.
1. The document discusses 4 problems involving limits, derivatives, and integrals. It provides the questions, solutions, and point values for each part.
2. The problems cover a range of calculus topics - finding limits, determining derivatives, solving differential equations, and evaluating definite integrals.
3. Overall, the document presents 4 multi-part problems that require applying core calculus concepts, with the goal of summarizing essential information at a high level in 3 sentences or less.
This document discusses how to sketch graphs resulting from transformations of basic parent functions, including horizontal and vertical stretches as well as translations. It provides examples of stretching and translating graphs of f(x) = x^2 and generalizes the effects of stretches and translations on basic parent functions. The document concludes with instructions on combining multiple transformations and an example problem determining the x-intercepts resulting from a composite transformation.
The document provides solutions to tutorial problems on differential geometry. It first shows that the differential of a function from a surface to 3D space is linear. It then calculates the Gauss map, Weingarten map, and principal curvatures for a sphere, surface of revolution, and other surfaces. The solutions involve parametrizing the surfaces and computing derivatives of the parametrizations.
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).
1) The volume of the solid region bounded by z = 9 - x^2 - y in the first octant is found using iterated integration.
2) The volume of the region bounded by z = x^2 + y^2, x^2 + y^2 = 25, and the xy-plane is found using polar coordinates.
3) The double integral of sin(x^2) over the region from 0 to 9 in x and y from 0 to x is evaluated.
This document contains a review of graph transformations including translations, reflections, stretches, and compressions. It provides examples of transforming the graphs of functions by sketching the original graph and the images resulting from various combinations of transformations. It also includes determining equations to describe the transformed graphs.
X2 T04 06 curve sketching - roots of functionsNigel Simmons
The document discusses how to sketch the graph of y = f(x). It notes that f(x) must be greater than or equal to 0 within its domain. The graph passes the x-axis where f(x) = 0. Critical points occur where the derivative f'(x) is 0. The graph has the same general shape as y = f(x) but is reflected above or below the line y = x depending on if f(x) is greater than or less than 1.
This document contains mathematical expressions and equations including: terms with variables a, b, c and x; logarithmic and exponential functions; integrals; and matrices. It appears to be showing various mathematical concepts and operations without additional context or explanation.
This document contains mathematical expressions and equations including: terms with variables a, b, c and x; logarithmic and exponential functions; integrals; and matrices. It appears to be showing various mathematical concepts and operations without additional context or explanation.
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
The document discusses calculating the area below the x-axis (A) for different functions f(x). It shows that A is given by the integral of f(x) from the left bound to the right bound, or equivalently the negative integral from the right bound to the left bound. As an example, it calculates A for the function f(x)=x^3 from -1 to 1, showing A = 1/2. It also notes that for odd functions, A can be calculated as half the integral from 0 to 1.
The document discusses how to sketch the graph of functions of the form y = f(x). It notes that f(x) must be greater than or equal to 0 within its domain. The graph of y = f(x) can be drawn and critical points occur where f(x) = 0 or f'(x) = 0. The shape of the graph is determined by whether f(x) is greater than or less than 1.
The document discusses the sine rule and its applications in solving problems involving triangles. It introduces the sine rule formula relating the ratios of sides and opposite angles. Examples are provided to demonstrate using the sine rule to calculate unknown side lengths and angles. It also covers calculating the area of a triangle using various formulas relating the area to combinations of sides and angles.
A geometric series is a sequence where each term is found by multiplying the previous term by a constant called the common ratio. The document defines the common ratio r and provides the general formula for calculating any term Tn in a geometric series. It also gives examples of (i) finding r and the general term for the series 2, 8, 32,... and (ii) finding r if T2=7 and T4=49. Finally, it solves (iii) finding the first term greater than 500 for the series 1, 4, 16,...
12 x1 t08 06 binomial probability (2012)Nigel Simmons
The document discusses binomial probability distributions. It explains that if an event has two possible outcomes and is repeated, the probability of each outcome follows a binomial distribution. It provides examples of calculating binomial probabilities for 1, 2, 3, and 4 events. The key points are:
- Binomial probabilities use the formula P(X=k) = nCk * pk * (1-p)(n-k)
- Where X is the number of successes, n is the number of trials, p is the probability of success on each trial, and q is the probability of failure.
- It illustrates calculating binomial probabilities for examples like drawing balls from a bag.
The document outlines the steps of mathematical induction to prove that 2n is greater than n^2 for all integers n greater than 4. It shows:
1) The basis step that proves this is true for n=5
2) The induction assumption that the statement holds true for some integer k greater than 4
3) The induction step that proves the statement holds true for k+1 if it is true for k
4) By the principle of mathematical induction, the statement must be true for all integers greater than 4.
The document discusses the concept of slope (gradient) of a line. It provides three definitions of slope: (1) the vertical rise over the horizontal run, (2) the change in y-values over the change in x-values between two points, and (3) the tangent of the angle of inclination. It also discusses properties of parallel and perpendicular lines, defining that parallel lines have the same slope while perpendicular lines have slopes that are negative reciprocals of each other. An example problem finds the value of a that would make two lines perpendicular.
The document discusses calculating the area below the x-axis (A) for different functions f(x). It shows that A is given by the integral of f(x) from the left bound to the right bound, or equivalently the negative integral from the right bound to the left bound. As an example, it calculates A for the function f(x)=x^3 from -1 to 1, showing A = 1/2. It also notes that for odd functions, A can be calculated as half the integral from 0 to 1.
The document proves the alternate segment theorem, which states that the angle formed by a tangent to a circle and a chord drawn to the point of contact is equal to any angle in the alternate segment. It does this by joining various points on the circle with lines, using properties of angles on tangents, chords, and in segments. It concludes that the angles ABP and APY are equal, proving the theorem.
1. The document provides solutions to trigonometric and algebraic equations. It solves equations involving sin, cos, tan functions as well as polynomials with variables x, y, z.
2. The algebraic equations section involves solving polynomials for single variables x or y, as well as systems of equations with variables x and y.
3. The trigonometric equations section expresses the general solutions to equations involving sin, cos, and tan functions in terms of radian angles n*pi/b, where b is the denominator and n is any integer.
1. The document discusses 4 problems involving limits, derivatives, and integrals. It provides the questions, solutions, and point values for each part.
2. The problems cover a range of calculus topics - finding limits, determining derivatives, solving differential equations, and evaluating definite integrals.
3. Overall, the document presents 4 multi-part problems that require applying core calculus concepts, with the goal of summarizing essential information at a high level in 3 sentences or less.
This document discusses how to sketch graphs resulting from transformations of basic parent functions, including horizontal and vertical stretches as well as translations. It provides examples of stretching and translating graphs of f(x) = x^2 and generalizes the effects of stretches and translations on basic parent functions. The document concludes with instructions on combining multiple transformations and an example problem determining the x-intercepts resulting from a composite transformation.
The document provides solutions to tutorial problems on differential geometry. It first shows that the differential of a function from a surface to 3D space is linear. It then calculates the Gauss map, Weingarten map, and principal curvatures for a sphere, surface of revolution, and other surfaces. The solutions involve parametrizing the surfaces and computing derivatives of the parametrizations.
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).
1) The volume of the solid region bounded by z = 9 - x^2 - y in the first octant is found using iterated integration.
2) The volume of the region bounded by z = x^2 + y^2, x^2 + y^2 = 25, and the xy-plane is found using polar coordinates.
3) The double integral of sin(x^2) over the region from 0 to 9 in x and y from 0 to x is evaluated.
This document contains a review of graph transformations including translations, reflections, stretches, and compressions. It provides examples of transforming the graphs of functions by sketching the original graph and the images resulting from various combinations of transformations. It also includes determining equations to describe the transformed graphs.
X2 T04 06 curve sketching - roots of functionsNigel Simmons
The document discusses how to sketch the graph of y = f(x). It notes that f(x) must be greater than or equal to 0 within its domain. The graph passes the x-axis where f(x) = 0. Critical points occur where the derivative f'(x) is 0. The graph has the same general shape as y = f(x) but is reflected above or below the line y = x depending on if f(x) is greater than or less than 1.
This document contains mathematical expressions and equations including: terms with variables a, b, c and x; logarithmic and exponential functions; integrals; and matrices. It appears to be showing various mathematical concepts and operations without additional context or explanation.
This document contains mathematical expressions and equations including: terms with variables a, b, c and x; logarithmic and exponential functions; integrals; and matrices. It appears to be showing various mathematical concepts and operations without additional context or explanation.
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
The document discusses calculating the area below the x-axis (A) for different functions f(x). It shows that A is given by the integral of f(x) from the left bound to the right bound, or equivalently the negative integral from the right bound to the left bound. As an example, it calculates A for the function f(x)=x^3 from -1 to 1, showing A = 1/2. It also notes that for odd functions, A can be calculated as half the integral from 0 to 1.
The document discusses how to sketch the graph of functions of the form y = f(x). It notes that f(x) must be greater than or equal to 0 within its domain. The graph of y = f(x) can be drawn and critical points occur where f(x) = 0 or f'(x) = 0. The shape of the graph is determined by whether f(x) is greater than or less than 1.
The document discusses the sine rule and its applications in solving problems involving triangles. It introduces the sine rule formula relating the ratios of sides and opposite angles. Examples are provided to demonstrate using the sine rule to calculate unknown side lengths and angles. It also covers calculating the area of a triangle using various formulas relating the area to combinations of sides and angles.
A geometric series is a sequence where each term is found by multiplying the previous term by a constant called the common ratio. The document defines the common ratio r and provides the general formula for calculating any term Tn in a geometric series. It also gives examples of (i) finding r and the general term for the series 2, 8, 32,... and (ii) finding r if T2=7 and T4=49. Finally, it solves (iii) finding the first term greater than 500 for the series 1, 4, 16,...
12 x1 t08 06 binomial probability (2012)Nigel Simmons
The document discusses binomial probability distributions. It explains that if an event has two possible outcomes and is repeated, the probability of each outcome follows a binomial distribution. It provides examples of calculating binomial probabilities for 1, 2, 3, and 4 events. The key points are:
- Binomial probabilities use the formula P(X=k) = nCk * pk * (1-p)(n-k)
- Where X is the number of successes, n is the number of trials, p is the probability of success on each trial, and q is the probability of failure.
- It illustrates calculating binomial probabilities for examples like drawing balls from a bag.
The document outlines the steps of mathematical induction to prove that 2n is greater than n^2 for all integers n greater than 4. It shows:
1) The basis step that proves this is true for n=5
2) The induction assumption that the statement holds true for some integer k greater than 4
3) The induction step that proves the statement holds true for k+1 if it is true for k
4) By the principle of mathematical induction, the statement must be true for all integers greater than 4.
The document discusses the concept of slope (gradient) of a line. It provides three definitions of slope: (1) the vertical rise over the horizontal run, (2) the change in y-values over the change in x-values between two points, and (3) the tangent of the angle of inclination. It also discusses properties of parallel and perpendicular lines, defining that parallel lines have the same slope while perpendicular lines have slopes that are negative reciprocals of each other. An example problem finds the value of a that would make two lines perpendicular.
The document discusses calculating the area below the x-axis (A) for different functions f(x). It shows that A is given by the integral of f(x) from the left bound to the right bound, or equivalently the negative integral from the right bound to the left bound. As an example, it calculates A for the function f(x)=x^3 from -1 to 1, showing A = 1/2. It also notes that for odd functions, A can be calculated as half the integral from 0 to 1.
The document proves the alternate segment theorem, which states that the angle formed by a tangent to a circle and a chord drawn to the point of contact is equal to any angle in the alternate segment. It does this by joining various points on the circle with lines, using properties of angles on tangents, chords, and in segments. It concludes that the angles ABP and APY are equal, proving the theorem.
Newton's Laws of Motion describe the relationship between an object's motion and the forces acting on it. The three laws are:
1. An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
2. The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
3. For every action, there is an equal and opposite reaction.
11 x1 t03 05 regions in the plane (2012)Nigel Simmons
The document provides steps for drawing regions defined by inequalities in the plane:
1. Use dotted lines for < or > and solid lines for ≤ or ≥.
2. Circle points of intersection that are not included.
3. Test points not on the curve to determine if they satisfy the inequalities and are within the region.
Examples are worked through, finding regions defined by y < x + 3, x2 + y2 ≥ 9, y ≤ 4 - x2, and the intersection of y ≥ x2 and y ≤ 3x + 4.
The document discusses real numbers and their properties. It covers prime factorization, finding the highest common factor (HCF), finding the lowest common multiple (LCM), and divisibility tests. Prime factorization represents a number as the product of its prime factors. To find the HCF, write numbers as products of their prime factors and take the common factors. To find the LCM, write numbers as products of their prime factors and take the factors using the highest powers. Divisibility tests provide shortcuts to determine if a number is divisible by 2, 3, 4, 5, 6, or 7.
The document discusses permutations of objects where some objects are the same. It provides examples of calculating permutations for sets with different numbers of objects (2, 3, 4 objects) and where some objects are the same. The key points are:
- To calculate permutations when some objects are the same, you divide the total permutations by the permutations of the identical objects
- Examples are provided calculating permutations for word arrangements giving different numbers based on identical letters
- The document also provides the general formula for calculating permutations when some objects are identical as n! / x! where n is the total objects and x is the number of identical objects.
11 x1 t13 07 products of intercepts (2013)Nigel Simmons
The document discusses products of intercepts for chords and secants of a circle. It states that the product of the intercept formed by a chord and one point (AX) and the intercept formed by the same chord and another point (BX) is equal to the product of the intercepts formed by another chord and the two points (CX and DX). It also notes that for secants, which intersect outside the circle, the same relationship holds for the intercepts formed. Finally, it states that the product of the intercepts formed by tangents from a point is equal to the square of the radius.
This document outlines three converse theorems related to cyclic quadrilaterals:
(1) If a diameter of a circle is the hypotenuse of a right triangle, the third vertex lies on the circle.
(2) If two points subtend the same angle from an interval on the same side, the four points are concyclic.
(3) If a quadrilateral has a pair of supplementary opposite angles or an exterior angle equals an interior opposite angle, the quadrilateral is cyclic.
The document discusses the primitive function and how it can be used to find the original curve when the equation of its tangent is known. It provides examples of calculating primitive functions from equations of tangents. It also gives an example of finding the equation of a curve given its point of intersection and gradient function.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example problem demonstrating these steps, eliminating variables through multiplication and addition of the equations until a single variable remains that can be solved for. The document notes that simultaneous equations with the same number of variables and equations can always be solved using this method.
The document defines functions, their domains and ranges, and properties of functions such as one-to-one, onto, and inverse functions. It also discusses the pigeonhole principle and its application to functions. Key concepts covered include function notation, domain and range, one-to-one and onto functions, inverse functions, and the generalized pigeonhole principle applied to functions from one set to another.
1. The document defines several functions and their domains and ranges. It also defines function compositions.
2. An example function is defined as f(x) = 2x and another is defined as g(x) = x + 1. It is shown that these functions are equal.
3. Several other example functions are defined, including trigonometric, polynomial, and rational functions.
4. Function compositions are defined for specific functions f and g over the domain of positive integers, and examples are given to illustrate function composition.
This document provides questions about finding the equations of tangents and normals to various polynomial functions at given points. It contains 8 parts with multiple questions each about finding the equations of tangents to polynomial curves and normals to polynomial curves at specified points using differentiation.
The document defines and provides examples of different types of functions:
1. Constant functions where f(x) = a for all values of x (e.g. f(x) = 2).
2. Linear functions of the form f(x) = ax + b (e.g. f(x) = 5x+3).
3. Quadratic functions of the form f(x) = ax2 + bx + c (e.g. f(x) = 3x2 + 2x + 1).
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
1) The document defines properties of exponential and logarithmic functions including: exponential functions follow exponent laws, logarithmic functions follow logarithmic laws, and the derivatives of exponentials and logarithms are the exponential/logarithmic functions themselves multiplied by the exponent/logarithm's argument.
2) Rules for limits of exponentials and logarithms as the argument approaches positive/negative infinity or zero are provided.
3) Graphs of the natural logarithm and logarithms with base a > 1 are similar shapes that increase without bound as the argument increases from 0 to infinity.
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
This document discusses quadratic functions and their maxima and minima. It provides the standard form and vertex form of quadratic equations, and explains how to find the vertex and y-intercept of a quadratic function. It derives the vertex formula, stating that the maximum or minimum of a quadratic occurs at the x-coordinate of the vertex. Finally, it provides homework problems involving finding maxima, minima, vertices, and y-intercepts of various quadratic functions.
1. The document provides graphs and equations for functions f(x), g(x), and their combinations. It asks the reader to sketch graphs, determine domains and ranges, and solve related problems.
2. The key combinations are addition, subtraction, multiplication, and division of f(x) and g(x). Their domains and ranges are identified from the original function graphs.
3. For a combination like f(x) + g(x), the domain is the same as the more restrictive of the two original functions, while the range includes all outputs equal to or greater than the original function ranges.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths in different triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
This document contains a summary of key concepts in algebra, geometry, and trigonometry:
1) Algebra topics include arithmetic operations, factoring, exponents, binomials, and the quadratic formula.
2) Geometry topics cover lines, triangles, circles, spheres, cones, cylinders, sectors, and trapezoids including formulas for area, perimeter, volume, and surface area.
3) Trigonometry definitions and formulas are provided for sine, cosine, tangent, cotangent, addition, subtraction, and half-angle identities.
This document contains a summary of key concepts in algebra, geometry, and trigonometry:
1) Algebra topics include arithmetic operations, factoring, exponents, binomials, and the quadratic formula.
2) Geometry topics cover lines, triangles, circles, spheres, cones, cylinders, sectors, and trapezoids including formulas for area, perimeter, volume, and surface area.
3) Trigonometry definitions and formulas are provided for sine, cosine, tangent, cotangent, addition, subtraction, and half-angle identities.
11 x1 t01 08 completing the square (2013)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows examples of solving equations in the form (i) x^2 + bx + c = 0, (ii) ax^2 + bx + c = 0, and (iii) x^2 - 6x + 6 = 0. The method involves grouping like terms and factorizing the equation into the form (x + p)^2 = q to extract the solutions.
The document defines the derivative and discusses rules for computing derivatives. It introduces the derivative as describing the slope of a curve at a point. It then outlines several basic rules for determining derivatives, such as the power rule, sum rule, and rules for constants and combinations of functions. The document also discusses the product rule, chain rule, and applications of derivatives to motion and rates of change problems.
This document contains 23 multiple choice questions related to mathematics and statistics. The questions cover topics such as logic, sets, functions, probability, geometry, and number theory. For each question, 4 possible answer choices are provided.
The document contains mathematical formulas and sets. It defines several sets including sets A and B and discusses their properties and relationships. It considers concepts like subsets, membership, cardinality, and power sets. Several examples are provided to illustrate set operations and properties.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
6. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x) ba
A f a f b
2
y y = f(x)
a b x
ca bc
A f a f c f c f b
2 2
a c b x
7. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x) ba
A f a f b
2
y y = f(x)
a b x
ca bc
A f a f c f c f b
2 2
ca
f a 2 f c f b
2
a c b x
10. y
y = f(x)
ca d c
A f a f c f c f d
2 2
bd
f d f b
2
a c d b x
11. y
y = f(x)
ca d c
A f a f c f c f d
2 2
bd
f d f b
2
a c d b x c a f a 2 f c 2 f d f b
2
12. y
y = f(x)
ca d c
A f a f c f c f d
2 2
bd
f d f b
2
a c d b x c a f a 2 f c 2 f d f b
2
In general;
13. y
y = f(x)
ca d c
A f a f c f c f d
2 2
bd
f d f b
2
a c d b x c a f a 2 f c 2 f d f b
2
In general; b
Area f x dx
a
14. y
y = f(x)
ca d c
A f a f c f c f d
2 2
bd
f d f b
2
a c d b x c a f a 2 f c 2 f d f b
2
In general; b
Area f x dx
a
h
y0 2 yothers yn
2
15. y
y = f(x)
ca d c
A f a f c f c f d
2 2
bd
f d f b
2
a c d b x c a f a 2 f c 2 f d f b
2
In general; b
Area f x dx
a
h
y0 2 yothers yn
2
ba
where h
n
n number of trapeziums
16. y
y = f(x)
ca d c
A f a f c f c f d
2 2
bd
f d f b
2
a c d b x c a f a 2 f c 2 f d f b
2
In general; b
Area f x dx
a
h
y0 2 yothers yn NOTE: there is
2
ba always one more
where h function value
n
than interval
n number of trapeziums
17. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
18. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba
h
n
20
4
0.5
19. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba
h
n x 0 0.5 1 1.5 2
20 y 2 1.9365 1.7321 1.3229 0
4
0.5
20. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba
h
n x 0 0.5 1 1.5 2
20 y 2 1.9365 1.7321 1.3229 0
h
4 Area y0 2 yothers yn
0.5 2
21. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba 1 1
h
n x 0 0.5 1 1.5 2
20 y 2 1.9365 1.7321 1.3229 0
h
4 Area y0 2 yothers yn
0.5 2
22. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba 1 2 2 2 1
h
n x 0 0.5 1 1.5 2
20 y 2 1.9365 1.7321 1.3229 0
h
4 Area y0 2 yothers yn
0.5 2
23. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba 1 2 2 2 1
h
n x 0 0.5 1 1.5 2
20 y 2 1.9365 1.7321 1.3229 0
h
4 Area y0 2 yothers yn
0.5 2
0.5
2 21.9365 1.7321 1.3229 0
2
2.996 units 2
24. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba 1 2 2 2 1
h
n x 0 0.5 1 1.5 2
20 y 2 1.9365 1.7321 1.3229 0
h
4 Area y0 2 yothers yn
0.5 2
0.5
2 21.9365 1.7321 1.3229 0
2
2.996 units 2 exact value π
25. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y 4 x , between x 0 and x 2
1
2 2
correct to 3 decimal points
ba 1 2 2 2 1
h
n x 0 0.5 1 1.5 2
20 y 2 1.9365 1.7321 1.3229 0
h
4 Area y0 2 yothers yn
0.5 2
0.5
2 21.9365 1.7321 1.3229 0
2
2.996 units 2 exact value π
3.142 2.996
% error 100
3.142
4.6%
28. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
29. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
30. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
e.g.
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
31. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
e.g.
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
h
Area y0 4 yodd 2 yeven yn
3
32. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
e.g. 1 1
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
h
Area y0 4 yodd 2 yeven yn
3
33. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
e.g. 1 4 4 1
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
h
Area y0 4 yodd 2 yeven yn
3
34. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
e.g. 1 4 2 4 1
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
h
Area y0 4 yodd 2 yeven yn
3
35. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
e.g. 1 4 2 4 1
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
h
Area y0 4 yodd 2 yeven yn
3
0.5
2 41.9365 1.3229 21.7321 0
3
3.084 units 2
36. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
where h
n
n number of intervals
e.g. 1 4 2 4 1
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
h
Area y0 4 yodd 2 yeven yn
3
0.5
2 41.9365 1.3229 21.7321 0 3.142 3.084
3 % error 100
3.084 units 2 3.142
1.8%