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CalculusCalculus: Early Transcendental Functions (MindTap Course List)6th Edition
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WRITING ABOUT CONCEPTS Finding a Particular Solution In Exercises 35–42, find the particular solution that satisfies the differential equation and the initial condition.

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Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770

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Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770
Chapter 5.1, Problem 48E
Textbook Problem
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WRITING ABOUT CONCEPTS

Finding a Particular Solution In Exercises 35–42, find the particular solution that satisfies the differential equation and the initial condition.

To determine

Your observation about the derivatives of f(x) and g(x) and relationship between f(x) and g(x).

Explanation of Solution

Given:

f(x)=tan2x and g(x)=sec2x

We have,

f(x)=tan2x.....(1)

 g(x)=sec2x...(2)

Differentiating equation (1) with respect to x, we get,

f′(x)=ddx(tan2x)=2tanx(sec2x)[ddxtanx=sec2x]f′(x)=2sec2xtanx......(3)

Differentiatingequation (2) with respect to x, we get,

g′(x)=ddx(sec2x)=2secx(secxtanx)[ddxsecx=secxtanx]g′(x)=2sec2xtanx

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Calculus: Early Transcendental Functions (MindTap Course List)
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Ch. 5.1 - Integration and Differentiation In Exercises 5 and...Ch. 5.1 - Integration and Differentiation In exercises 5 and...Ch. 5.1 - Solving a Differential Equation In Exercises 7-10,...Ch. 5.1 - Solving a Differential Equation In Exercises 7-10,...Ch. 5.1 - Solving a Differential Equation In Exercises 7-10,...Ch. 5.1 - (i) Find the general solution of the differential...Ch. 5.1 - Rewriting Before Integrating In Exercises 11-14,...Ch. 5.1 - Complete the following table to find the...Ch. 5.1 - Rewriting before Integrating in Exercises 11-14,...Ch. 5.1 - Rewriting Before Integrating In Exercises 11-14,...
Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 1132,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - (i) Find the indefinite integral (x+12x)dx (ii)...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 1132,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 1132,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 1132,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 1132,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Finding an Indefinite Integral In Exercises 15-36,...Ch. 5.1 - Sketching a Graph In Exercises 33 and 34, the...Ch. 5.1 - Sketching a Graph In Exercises 33 and 34, the...Ch. 5.1 - Finding a Particular Solution In Exercises 37-44,...Ch. 5.1 - Finding a Particular Solution In Exercises 3542,...Ch. 5.1 - Finding a Particular Solution In Exercises 37-44,...Ch. 5.1 - Finding a Particular Solution In Exercises 3542,...Ch. 5.1 - Finding a Particular Solution In Exercises 37-44,...Ch. 5.1 - Finding a Particular Solution In Exercises 37-44,...Ch. 5.1 - Finding a Particular Solution In Exercises 37-44,...Ch. 5.1 - Finding a Particular Solution In Exercises 37-44,...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - WRITING ABOUT CONCEPTS Antiderivatives and...Ch. 5.1 - WRITING ABOUT CONCEPTS Finding a Particular...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66. find the...Ch. 5.1 - Finding an Equation of a Tangent Line In exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Horizontal Tangent Find a function f such that the...Ch. 5.1 - Finding a FunctionThe graph of f' is shown. 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In Exercises 69-74, determine...Ch. 5.3 - True or False? In Exercises 69-74, determine...Ch. 5.3 - True or False? In Exercises 69-74, determine...Ch. 5.3 - Finding a Riemann Sum Find the Riemann sum for...Ch. 5.3 - Finding a Riemann Sum Find the Riemann sum for...Ch. 5.3 - Proof prove that abxdx=b2a22.Ch. 5.3 - Proof prove that abx2dx=b3a33.Ch. 5.3 - Think About It Determine whether the Dirichlet...Ch. 5.3 - Finding a Definite Integral The function...Ch. 5.3 - Finding Values Find the constants a and b that...Ch. 5.3 - Step Function Evaluate, if possible, the integral...Ch. 5.3 - Using a Riemann Sum Determine...Ch. 5.3 - For each continuous function f:[0,1],let...Ch. 5.4 - Graphical Reasoning In Exercises 5-8, use a...Ch. 5.4 - Graphical Reasoning In Exercises 5-8, use a...Ch. 5.4 - Graphical Reasoning In Exercises 5-8, use a...Ch. 5.4 - Graphical Reasoning In Exercises 5-8, use a...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 538,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Evaluating a Definite Integral In Exercises 9-34,...Ch. 5.4 - Finding the Area of a Region In Exercises 35-38,...Ch. 5.4 - Finding the Area of a Region In Exercises 35-38,...Ch. 5.4 - Finding the Area of a Region In Exercises 35-38,...Ch. 5.4 - Finding the Area of a Region In Exercises 35-38,...Ch. 5.4 - Finding the Area of a Region In Exercises 39-44,...Ch. 5.4 - Finding the Area of a Region In Exercises 39-44,...Ch. 5.4 - Finding the Area of a Region In Exercises 39-44,...Ch. 5.4 - Finding the Area of a Region In Exercises 39-44,...Ch. 5.4 - Finding the Area of a Region In Exercises 39-44,...Ch. 5.4 - Finding the Area of a Region In Exercises 39-44,...Ch. 5.4 - Using the Mean Value Theorem for Integrals In...Ch. 5.4 - Using the Mean Value Theorem for Integrals In...Ch. 5.4 - Using the Mean Value Theorem for Integrals In...Ch. 5.4 - Using the Mean Value Theorem for Integrals In...Ch. 5.4 - Using the Mean Value Theorem for Integrals In...Ch. 5.4 - Using the Mean Value Theorem for Integrals In...Ch. 5.4 - Finding the Average Value of a Function In...Ch. 5.4 - Finding the Average Value of a Function In...Ch. 5.4 - Finding the Average Value of a Function In...Ch. 5.4 - Finding the Average Value of a Function In...Ch. 5.4 - Finding the Average Value of a Function In...Ch. 5.4 - Finding the Average Value of a Function In...Ch. 5.4 - Using a Graph The graph of f is shown in the...Ch. 5.4 - Rate of Growth Let r'(t) represent the rate of...Ch. 5.4 - Force The force F (in newtons) of a hydraulic...Ch. 5.4 - Blood Flow The velocity v of the flow of blood at...Ch. 5.4 - Button's Needle Experiment A horizontal plane is...Ch. 5.4 - HOW DO YOU SEE IT? 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In Exercises 109 and 110, determine...Ch. 5.4 - True or False? In Exercises 117 and 118, determine...Ch. 5.4 - Analyzing a Function Show that the function...Ch. 5.4 - Finding a Function Find the function f(x) and all...Ch. 5.4 - Finding Values Let G(x)=0x[s0sf(t)dt]ds where f is...Ch. 5.4 - Proof Prove that...Ch. 5.5 - Finding uand du In Exercises 14, complete the...Ch. 5.5 - Recognizing Patterns In Exercises 5-8, complete...Ch. 5.5 - Recognizing Patterns In Exercises 5-8, complete...Ch. 5.5 - Recognizing Patterns In Exercises 5-8, complete...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 526,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 526,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 526,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 526,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 526,...Ch. 5.5 - x2(1+x3)2dxCh. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 526,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 9-28,...Ch. 5.5 - Differential Equation In Exercises 29-32, find the...Ch. 5.5 - Differential Equation In Exercises 29-32, find the...Ch. 5.5 - Differential Equation In Exercises 29-32, find the...Ch. 5.5 - Differential Equation In Exercises 2730, solve the...Ch. 5.5 - Slope Field In Exercises 33 and 34, a differential...Ch. 5.5 - Slope Field In Exercises 33 and 34, a differential...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 35-56,...Ch. 5.5 - Finding an Equation In Exercises 57-64, find an...Ch. 5.5 - Finding an Equation In Exercises 57-64, find an...Ch. 5.5 - Finding an Equation In Exercises 57-64, find an...Ch. 5.5 - Finding an Equation In Exercises 57-64, find an...Ch. 5.5 - Finding an Equation In Exercises 5560, find an...Ch. 5.5 - Finding an Equation In Exercises 5560, find an...Ch. 5.5 - Change of Variables In Exercises 65-72, find the...Ch. 5.5 - Change of Variables In Exercises 65-72, find the...Ch. 5.5 - Change of Variables In Exercises 65-72, find the...Ch. 5.5 - Change of Variables In Exercises 65-72, find the...Ch. 5.5 - Change of Variables In Exercises 65-72, find the...Ch. 5.5 - Change of Variables In Exercises 65-72, find the...Ch. 5.5 - Change of Variables In Exercises 6168, find the...Ch. 5.5 - Change of Variables In Exercises 6168, find the...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 6980,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 6980,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 73-84,...Ch. 5.5 - Finding the Area of a Region In Exercises 85-88,...Ch. 5.5 - Finding the Area of a Region In Exercises 85-88,...Ch. 5.5 - Finding the Area of a Region In Exercises 85-88,...Ch. 5.5 - Finding the Area of a Region In Exercises 85-88,...Ch. 5.5 - Area In Exercises 89-92, find the area of the...Ch. 5.5 - Area In Exercises 89-92, find the area of the...Ch. 5.5 - Area In Exercises 89-92, find the area of the...Ch. 5.5 - Area In Exercises 89-92, find the area of the...Ch. 5.5 - Even and Odd Functions In Exercises 93-96,...Ch. 5.5 - Even and Odd Functions In Exercises 93-96,...Ch. 5.5 - Even and Odd Functions In Exercises 93-96,...Ch. 5.5 - Even and Odd Functions In Exercises 93-96,...Ch. 5.5 - Using an Even Function Use 06x2dx=72 to evaluate...Ch. 5.5 - Using Symmetry Use the symmetry of the graphs of...Ch. 5.5 - Even and Odd Functions In Exercises 99 and 100,...Ch. 5.5 - Even and Odd Functions In Exercises 99 and 100,...Ch. 5.5 - WRITING ABOUT CONCEPTS Using Substitution Describe...Ch. 5.5 - WRITING ABOUT CONCEPTS Analyzing the Integrand...Ch. 5.5 - Even and Odd Functions In Exercises 99 and 100,...Ch. 5.5 - Even and Odd Functions In Exercises 99 and 100,...Ch. 5.5 - Depreciation The rate of depreciation dV/dt of a...Ch. 5.5 - HOW DO YOU SEE IT? The graph shows the flow rate...Ch. 5.5 - Sales The sales S (in thousands of units) of a...Ch. 5.5 - Electricity The oscillating current in an...Ch. 5.5 - Graphical Analysis Consider the functions f and g,...Ch. 5.5 - The probability that a person will remember...Ch. 5.5 - The probability that ore samples taken from a...Ch. 5.5 - Finding a Limit Using a Definite Integral Find...Ch. 5.5 - Rewriting Integrals (a) Show that...Ch. 5.5 - Rewriting Integral (a) Show that...Ch. 5.5 - True or False? In Exercises 111116, determine...Ch. 5.5 - True or False? In Exercises 113-118, determine...Ch. 5.5 - True or False? In Exercises 113-118, determine...Ch. 5.5 - True or False? In Exercises 113-118, determine...Ch. 5.5 - True or False? In Exercises 113-118, determine...Ch. 5.5 - True or False? In Exercises 113-118, determine...Ch. 5.5 - Rewriting Integrals Assume that f is continuous...Ch. 5.5 - Integration and Differentiation (a) Verify that...Ch. 5.5 - Proof Prove the second property of Theorem 5.16.Ch. 5.5 - Rewriting Integrals Show that if f is continuous...Ch. 5.5 - PUTNAM EXAM CHALLENGE If a0,a1,,an are real...Ch. 5.5 - PUTNAM EXAM CHALLENGE Find all the continuous...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule...Ch. 5.6 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5.6 - Polynomial Approximations The Trapezoidal Rule and...Ch. 5.6 - Describing an Error Describe the size of the error...Ch. 5.6 - Estimating Errors In Exercises 2528, use the error...Ch. 5.6 - Estimating Errors In Exercises 2528. use the error...Ch. 5.6 - Estimating Errors In Exercises 2528, use the error...Ch. 5.6 - Estimating Errors In Exercises 2528, use the error...Ch. 5.6 - Estimating Errors In Exercises 2932, use the error...Ch. 5.6 - Estimating Errors In Exercises 2932, use the error...Ch. 5.6 - Estimating Errors In Exercises 2932, use the error...Ch. 5.6 - Estimating Errors In Exercises 2932, use the error...Ch. 5.6 - Estimating Errors Using Technology In Exercises...Ch. 5.6 - Estimating Errors Using Technology In Exercises...Ch. 5.6 - Estimating Errors Using Technology In Exercises...Ch. 5.6 - Estimating Errors Using Technology In Exercises...Ch. 5.6 - Finding the Area of a Region Approximate the area...Ch. 5.6 - Finding the Area of a Region Approximate the area...Ch. 5.6 - Area Use Simpsons Rule with n = 14 to approximate...Ch. 5.6 - Circumference The elliptic...Ch. 5.6 - surveying Use the Trapezoidal Rule to estimate the...Ch. 5.6 - HOW DO YOU SEE IT? The function f(x) is concave...Ch. 5.6 - Work To determine the size of the motor required...Ch. 5.6 - Approximating a Function The table lists several...Ch. 5.6 - Approximation of Pi In Exercises 45 and 46, use...Ch. 5.6 - Approximation of Pi In Exercises 45 and 46, use...Ch. 5.6 - Using Simpson's Rule Use Simpsons Rule with n = 10...Ch. 5.6 - Proof Prove that Simpsons Rule is exact when...Ch. 5.6 - Proof Prove that you can find a polynomial...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 126,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 126,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 126,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 126,...Ch. 5.7 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.7 - Change of Variables In Exercises 29-32, find the...Ch. 5.7 - Finding an Indefinite Integral by u-Substitution...Ch. 5.7 - Change of Variables In Exercises 29-32, find the...Ch. 5.7 - Change of Variables In Exercises 29-32, find the...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.7 - Differential Equation In Exercises 43-46, find the...Ch. 5.7 - Differential Equation In Exercises 43-46, find the...Ch. 5.7 - Differential Equation In Exercises 43-46, find the...Ch. 5.7 - Differential Equation In Exercises 43-46, find the...Ch. 5.7 - Finding a Particular Solution In Exercises 47 and...Ch. 5.7 - Finding a Particular Solution In Exercises 47 and...Ch. 5.7 - Slope Field In Exercises 49 and 50, a differential...Ch. 5.7 - Slope Field In Exercises 49 and 50, a differential...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.7 - Finding an Integral Using Technology In Exercises...Ch. 5.7 - Finding an Integral Using Technology In Exercises...Ch. 5.7 - Using Technology to Find an Integral In Exercises...Ch. 5.7 - Using Technology to Find an Integral In Exercises...Ch. 5.7 - Using Technology to Find an Integral In Exercises...Ch. 5.7 - Using Technology to Find an Integral In Exercises...Ch. 5.7 - Finding a Derivative In Exercises 61-64, find F (...Ch. 5.7 - Finding a Derivative In Exercises 61-64, find F (...Ch. 5.7 - Finding a Derivative In Exercises 6568. find F(x)....Ch. 5.7 - Finding a Derivative In Exercises 6568. find F(x)....Ch. 5.7 - Area In Exercises 65-68, find the area of the...Ch. 5.7 - Area In Exercises 6972, find the area of the given...Ch. 5.7 - Area In Exercises 6972, find the area of the given...Ch. 5.7 - Area In Exercises 65-68, find the area of the...Ch. 5.7 - Area In Exercises 65-68, find the area of the...Ch. 5.7 - Area In Exercises 65-68, find the area of the...Ch. 5.7 - Area In Exercises 65-68, find the area of the...Ch. 5.7 - Area In Exercises 65-68, find the area of the...Ch. 5.7 - Area In Exercises 65-68, find the area of the...Ch. 5.7 - Finding the Average Value of a Function In...Ch. 5.7 - Finding the Average Value of a Function In...Ch. 5.7 - Finding the Average Value of a Function In...Ch. 5.7 - Numerical Integration In Exercises 7780, use the...Ch. 5.7 - Numerical Integration In Exercises 7780, use the...Ch. 5.7 - Numerical Integration In Exercises 7780, use the...Ch. 5.7 - Numerical Integration In Exercises 7780, use the...Ch. 5.7 - WRITING ABOUT CONCEPTS Choosing a Formula In...Ch. 5.7 - WRITING ABOUT CONCEPTS Choosing a Formula In...Ch. 5.7 - WRITING ABOUT CONCEPTS Choosing a Formula In...Ch. 5.7 - WRITING ABOUT CONCEPTS Choosing a Formula In...Ch. 5.7 - EXPLORING CONCERTS f(x)=secx,[0,1] (a) 6 (b) 6 (c)...Ch. 5.7 - EXPLORING CONCEPTS f(x)=2xx2+1,[0,4] (a) 3 (b) 7...Ch. 5.7 - Finding a Value Find a value of x such that...Ch. 5.7 - Finding a Value Find a value of x such that 1x1tdt...Ch. 5.7 - Proof Prove that cotudu=lnsinu+CCh. 5.7 - Proof Prove that cotudu=lncscu+cotu+CCh. 5.7 - Using Properties of Logarithms and Trigonometric...Ch. 5.7 - Using Properties of Logarithms and Trigonometric...Ch. 5.7 - Using Properties of Logarithms and Trigonometric...Ch. 5.7 - Using Properties of Logarithms and Trigonometric...Ch. 5.7 - Population Growth A population of bacteria P is...Ch. 5.7 - Sales The rate of change in sales S is inversely...Ch. 5.7 - Heat Transfer Find the time required for an object...Ch. 5.7 - Average Price The demand equation for a product is...Ch. 5.7 - Area and Slope Graph the function f(x)=x/(1+x2) on...Ch. 5.7 - HOW DO YOU SEE IT? 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Using the graph, which value...Ch. 5.8 - Slope Field In Exercises 55 and 56, a differential...Ch. 5.8 - Slope Field In Exercises 55 and 56, a differential...Ch. 5.8 - Slope Field In Exercises 57-60, use a graphing...Ch. 5.8 - Slope Field In Exercises 57-60, use a graphing...Ch. 5.8 - Slope Field In Exercises 57-60, use a graphing...Ch. 5.8 - Slope Field In Exercises 57-60, use a graphing...Ch. 5.8 - Differential Equation In Exercises 61 and 62, find...Ch. 5.8 - Differential Equation In Exercises 61 and 62, find...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area In Exercises 6166, find the area of the...Ch. 5.8 - Area In Exercises 6166, find the area of the...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area (a) Sketch the region whose area is...Ch. 5.8 - Approximating Pi (a) Show that 0141+x2dx= (b)...Ch. 5.8 - Investigation Consider the function...Ch. 5.8 - Comparing integrals Consider the integral 16xx2dx...Ch. 5.8 - True or False? 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In Exercises 7174, determine...Ch. 5.8 - Verifying an Integration Rule the rule by...Ch. 5.8 - Verifying an Integration Rule In Exercises 75-75,...Ch. 5.8 - Verifying an Integration Rule In Exercises 73-75,...Ch. 5.8 - Proof Graph y1=x1+x2,y2=arctanx, and y1=x, on [0,...Ch. 5.8 - Numerical Integration (a) Write an integral that...Ch. 5.8 - Vertical Motion An object is projected upward from...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 714, verify the...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 714, verify the...Ch. 5.9 - Finding Values of Hyperbolic Functions In...Ch. 5.9 - Finding Values of Hyperbolic Functions In...Ch. 5.9 - Finding a Limit In Exercises 21-24, And the limit....Ch. 5.9 - Finding a Limit In Exercises 21-24, find the limit...Ch. 5.9 - Finding a Limit In Exercises 1722, find the limit....Ch. 5.9 - Finding a Limit In Exercises 1722, find the limit....Ch. 5.9 - Finding a Limit In Exercises 21-24, find the limit...Ch. 5.9 - Finding a Limit In Exercises 21-24, find the...Ch. 5.9 - Finding a Derivative In Exercises 2332, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 2332, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 2332, find the...Ch. 5.9 - Finding a Derivative In Exercises 2332, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.9 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.9 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Catenary In Exercises 43 and 44, a model for a...Ch. 5.9 - Catenary In Exercises 43 and 44, a model for a...Ch. 5.9 - Finding an Indefinite Integral In Exercises 4354,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 4354,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 4354,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 4354,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 5560,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 5560,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 5560,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Comparing Functions Discuss several ways in which...Ch. 5.9 - Hyperbolic Functions Which hyperbolic functions...Ch. 5.9 - Comparing Derivative Formulas Which hyperbolic...Ch. 5.9 - HOW DO YOU SEE IT? Use the graphs of f and g shown...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 6574, find the...Ch. 5.9 - Finding a Derivative In Exercises 6574, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 6574, find the...Ch. 5.9 - Finding a Derivative In Exercises 6574, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Differential Equation In Exercises 8790, solve the...Ch. 5.9 - Differential Equation In Exercises 8790, solve the...Ch. 5.9 - Differential Equation In Exercises 87 and 88, find...Ch. 5.9 - Differential Equation In Exercises 87 and 88, find...Ch. 5.9 - Area In Exercises 89-92, find the area of the...Ch. 5.9 - Area In Exercises 89-92, find the area of the...Ch. 5.9 - Area In Exercises 89-92, find the area of the...Ch. 5.9 - Area In Exercises 89-92, find the area of the...Ch. 5.9 - Chemical Reactions Chemicals A and B combine in a...Ch. 5.9 - Vertical MotionAn object is dropped from a height...Ch. 5.9 - Tractrix Consider the equation of a tractrix...Ch. 5.9 - Tractrix Show that the boat in Example 5 is always...Ch. 5.9 - Proof Prove that tanh1x=12ln(1+x1x),1x1Ch. 5.9 - Proof Prove that sinh1t=ln(t+t2+1)Ch. 5.9 - Using a Right Triangle Show that...Ch. 5.9 - Integration Let x0 and b0. 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Evaluating a Definite Integral In Exercises 63-70,...Ch. 5 - Evaluating a Definite Integral In Exercises 63-70,...Ch. 5 - Evaluating a Definite Integral In Exercises 63-70,...Ch. 5 - Evaluating a Definite Integral In Exercises 63-70,...Ch. 5 - Evaluating a Definite Integral In Exercises 63-70,...Ch. 5 - Evaluating a Definite Integral In Exercises 63-70,...Ch. 5 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5 - Using the Trapezoidal Rule and Simpson's Rule In...Ch. 5 - Finding an Indefinite Integral In Exercises 79-84,...Ch. 5 - Finding an Indefinite Integral In Exercises 79-84,...Ch. 5 - Finding an Indefinite Integral In Exercises 79-84,...Ch. 5 - Finding an Indefinite Integral In Exercises 79-84,...Ch. 5 - Finding an Indefinite Integral In Exercises 79-84,...Ch. 5 - Finding an Indefinite Integral In Exercises 79-84,...Ch. 5 - Evaluating a Definite Integral In Exercises 85-88,...Ch. 5 - Evaluating a Definite Integral In Exercises 85-88,...Ch. 5 - Evaluating a Definite Integral In Exercises 85-88,...Ch. 5 - Evaluating a Definite Integral In Exercises 85-88,...Ch. 5 - Finding an Indefinite Integral In Exercises 89-94,...Ch. 5 - Finding an Indefinite Integral In Exercises 89-94,...Ch. 5 - Finding an Indefinite Integral In Exercises 89-94,...Ch. 5 - Finding an Indefinite Integral In Exercises 89-94,...Ch. 5 - Finding an Indefinite Integral In Exercises 89-94,...Ch. 5 - Finding an Indefinite Integral In Exercises 89-94,...Ch. 5 - Finding a Derivative In Exercises 95-98, find the...Ch. 5 - Finding a Derivative In Exercises 95-98, find the...Ch. 5 - Finding a Derivative In Exercises 95-98, find the...Ch. 5 - Finding a Derivative In Exercises 95-98, find the...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Using a Function Let L(x)=1x1tdt,x0. (a) Find...Ch. 5 - Parabolic Arch Archimedes showed that the area of...Ch. 5 - Using a Continuous Function Let f be continuous on...Ch. 5 - Fresnel Function The Fresnel function S is defined...Ch. 5 - Approximation The Two-Point Gaussian Quadrature...Ch. 5 - Extrema and Points of Inflection The graph of the...Ch. 5 - Falling Objects Galileo Galilei (1564-1642) stated...Ch. 5 - Use the techniques of this chapter to verify this...Ch. 5 - Use the techniques of this chapter to verify this...Ch. 5 - Riemann Sum Use an appropriate Riemann sum to...Ch. 5 - Riemann Sum Use an appropriate Riemann sum to...Ch. 5 - Proof Let f be integrable on [a, b] and 0mf(x)M...Ch. 5 - Velocity and Acceleration A car travels in a...Ch. 5 - Proof Prove that if f is a continuous function on...Ch. 5 - Verifying a Sum Verify that i=1ni2=n(n+1)(2n+1)6...Ch. 5 - Area Consider the three regions A, B. and C...Ch. 5 - Area Use integration by substitution to find the...Ch. 5 - Area Use integration by substitution to find the...Ch. 5 - Approximating a Function (a) Use a graphing...

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