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7th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337552516

Chapter 5.1, Problem 41E

Textbook Problem

Finding a Particular Solution In Exercises 37-44, find the particular solution of the differential equation that satisfies the initial condition(s).

Calculus: Early Transcendental Functions

Ch. 5.1 - CONCEPT CHECK Antiderivative What does it mean for...Ch. 5.1 - CONCEPT CHECK Logarithmic Properties What is the...Ch. 5.1 - CONCEPT CHECK The Number e How is the number e...Ch. 5.1 - CONCEPT CHECK Differentiation of Logarithmic...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 15-36,...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 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 - 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 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 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 - 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 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 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.2 - CONCEPT CHECK Sigma Notation What are the index of...Ch. 5.2 - CONCEPT CHECK Sums What is the value of n? (a)...Ch. 5.2 - CONCEPT CHECK Upper and Lower Sums In your own...Ch. 5.2 - CONCEPT CHECK Finding Area by the Limit Definition...Ch. 5.2 - Finding a Sum In Exercises 5-10, find the sum by...Ch. 5.2 - Finding a Sum In Exercises 5-10, find the sum by...Ch. 5.2 - Finding a Sum In Exercises 5-10, find the sum by...Ch. 5.2 - Finding a Sum In Exercises 5-10, find the sum by...Ch. 5.2 - Finding a Sum In Exercises 5-10, find the sum by...Ch. 5.2 - Finding a Sum In Exercises 5-10, find the sum by...Ch. 5.2 - Using Sigma Notation In Exercises 11-16, use sigma...Ch. 5.2 - Using Sigma Notation In Exercises 11-16, use sigma...Ch. 5.2 - Using Sigma Notation In Exercises 11-16, use sigma...Ch. 5.2 - Using Sigma Notation In Exercises 11-16, use sigma...Ch. 5.2 - Using Sigma Notation In Exercises 11-16, use sigma...Ch. 5.2 - Using Sigma Notation In Exercises 11-16, use sigma...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 17-24, use the...Ch. 5.2 - Evaluating a Sum In Exercises 25-28, use the...Ch. 5.2 - Evaluating a Sum In Exercises 25-28, use the...Ch. 5.2 - Evaluating a Sum In Exercises 25-28, use the...Ch. 5.2 - Evaluating a Sum In Exercises 25-28, use the...Ch. 5.2 - Approximating the Area of a Plane Region In...Ch. 5.2 - Approximating the Area of a Plane Region In...Ch. 5.2 - Approximating the Area of a Plane Region In...Ch. 5.2 - Approximating the Area of a Plane Region In...Ch. 5.2 - Approximating the Area of a Plane Region In...Ch. 5.2 - Approximating the Area of a Plane Region In...Ch. 5.2 - Using Upper and Lower Sums In Exercises 35 and 36,...Ch. 5.2 - Using Upper and Lower Sums In Exercises 35 and 36,...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Finding Upper and Lower Sums for a Region In...Ch. 5.2 - Numerical Reasoning Consider a triangle of area 2...Ch. 5.2 - Numerical Reasoning Consider a trapezoid of area 4...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Finding Area by the Limit Definition In Exercises...Ch. 5.2 - Approximating Area with the Midpoint Rule In...Ch. 5.2 - Approximating Area with the Midpoint Rule In...Ch. 5.2 - Approximating Area with the Midpoint Rule In...Ch. 5.2 - Approximating Area with the Midpoint Rule In...Ch. 5.2 - Approximating Area with the Midpoint Rule In...Ch. 5.2 - Approximating Area with the Midpoint Rule In...Ch. 5.2 - EXPLORING CONCEPTS Approximation Determine which...Ch. 5.2 - EXPLORING CONCEPTS Approximation A function is...Ch. 5.2 - EXPLORING CONCEPTS Midpoint Rule Explain why the...Ch. 5.2 - EXPLORING CONCEPTS Midpoint Rule Does the Midpoint...Ch. 5.2 - Graphical Reasoning Consider the region bounded by...Ch. 5.2 - HOW DO YOU SEE IT? The function shown in the graph...Ch. 5.2 - True or False? In Exercises 75 and 76, determine...Ch. 5.2 - True or False? In Exercises 75 and 76, determine...Ch. 5.2 - Writing Use the figure to write a short paragraph...Ch. 5.2 - Graphical Reasoning Consider an n-sided regular...Ch. 5.2 - Seating Capacity A teacher places n seats to form...Ch. 5.2 - Proof Prove each formula mathematical induction....Ch. 5.2 - PUTNAM EXAM CHALLENGE A dart, thrown at random,...Ch. 5.3 - CONCEPT CHECK Riemann Sum What does a Riemann sum...Ch. 5.3 - CONCEPT CHECK Definite Integral Explain how to...Ch. 5.3 - Evaluating a Limit In Exercises 3 and 4, use...Ch. 5.3 - Evaluating a Limit In Exercises 3 and 4, use...Ch. 5.3 - Evaluating a Definite Integral as a Limit In...Ch. 5.3 - Evaluating a Definite Integral as a Limit In...Ch. 5.3 - Evaluating a Definite Integral as a Limit In...Ch. 5.3 - Evaluating a Definite Integral as a Limit In...Ch. 5.3 - Evaluating a Definite Integral as a Limit In...Ch. 5.3 - Evaluating a Definite Integral as a Limit In...Ch. 5.3 - Writing a Limit as a Definite Integral In...Ch. 5.3 - Writing a Limit as a Definite Integral In...Ch. 5.3 - Writing a Limit as a Definite Integral In...Ch. 5.3 - Writing a Limit as a Definite Integral In...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Writing a Definite Integral In Exercises 15-26,...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Evaluating a Definite Integral Using a Geometric...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals In...Ch. 5.3 - Using Properties of Definite Integrals Given...Ch. 5.3 - Using Properties of Definite Integrals Given...Ch. 5.3 - Using Properties of Definite Integrals Given...Ch. 5.3 - Using Properties of Definite Integrals Given...Ch. 5.3 - Estimating a Definite Integral Use the table of...Ch. 5.3 - Estimating a Definite Integral Use the table of...Ch. 5.3 - Think About It The graph of f consists of line...Ch. 5.3 - Think About It The graph of f consists of line...Ch. 5.3 - Think About It Consider a function f that is...Ch. 5.3 - HOW DO YOU SEE IT? Use the figure to fill in the...Ch. 5.3 - Think About It A function f is defined below. Use...Ch. 5.3 - Think About It A function f is defined below. Use...Ch. 5.3 - EXPLORING CONCEPTS Approximation In Exercises...Ch. 5.3 - EXPLORING CONCEPTS Approximation In Exercises...Ch. 5.3 - EXPLORING CONCEPTS Approximation In Exercises...Ch. 5.3 - EXPLORING CONCEPTS Approximation In Exercises...Ch. 5.3 - Verifying a Rule Use a graph to explain why...Ch. 5.3 - Verifying a Property Use a graph to explain why...Ch. 5.3 - Using Different Methods Describe two ways to...Ch. 5.3 - Finding a Function Give an example of a function...Ch. 5.3 - Finding Values In Exercises 65-68, find possible...Ch. 5.3 - Finding Values In Exercises 65-68, find possible...Ch. 5.3 - Finding Values In Exercises 65-68, find possible...Ch. 5.3 - Finding Values In Exercises 65-68, find possible...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 - True or False? 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 - 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 - Finding Values Find the constants a and b where a ...Ch. 5.3 - Think About It When is abf(x)dx=ab|f(x)|dx?...Ch. 5.3 - Step Function Evaluate, if possible, the integral...Ch. 5.3 - Using a Riemann Sum Determine...Ch. 5.4 - CONCEPT CHECK Fundamental Theorem of Calculus...Ch. 5.4 - CONCEPT CHECK Mean Value Theorem Describe the Mean...Ch. 5.4 - CONCEPT CHECK Average Value of a Function Describe...Ch. 5.4 - CONCEPT CHECK Accumulation Function Why is...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 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 - 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 - 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 - Force The force F (in newtons) of a hydraulic...Ch. 5.4 - Respiratory Cycle The volume V, in liters, of air...Ch. 5.4 - Button's Needle Experiment A horizontal plane is...Ch. 5.4 - HOW DO YOU SEE IT? The graph of f is shown in the...Ch. 5.4 - Evaluating a Definite Integral In Exercises 61 and...Ch. 5.4 - Evaluating a Definite Integral In Exercises 61 and...Ch. 5.4 - Evaluating a Definite Integral In Exercises 63 and...Ch. 5.4 - Evaluating a Definite Integral In Exercises 63 and...Ch. 5.4 - Analyzing a Function Let g(x)=0xf(t)dt, where f is...Ch. 5.4 - Analyzing a Function Let g(x)=0xf(t)dt, where f is...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Finding and Checking an Integral In Exercises...Ch. 5.4 - Using the Second Fundamental Theorem of Calculus...Ch. 5.4 - Using the Second Fundamental Theorem of Calculus...Ch. 5.4 - Using the Second Fundamental Theorem of Calculus...Ch. 5.4 - Using the Second Fundamental Theorem of Calculus...Ch. 5.4 - Using the Second Fundamental Theorem of Calculus...Ch. 5.4 - Using the Second Fundamental Theorem of Calculus...Ch. 5.4 - Finding a Derivative In Exercises 81-86, find...Ch. 5.4 - Finding a Derivative In Exercises 81-86, find...Ch. 5.4 - Finding a Derivative In Exercises 81-86, find...Ch. 5.4 - Finding a Derivative In Exercises 81-86, find...Ch. 5.4 - Finding a Derivative In Exercises 81-86, find...Ch. 5.4 - Finding a Derivative In Exercises 81-86, find...Ch. 5.4 - Graphical Analysis Sketch an approximate graph or...Ch. 5.4 - Area The area A between the graph of the function...Ch. 5.4 - Water Flow Water flows from a storage tank at a...Ch. 5.4 - Oil Leak At 1:00 p.m., oil begins leaking from a...Ch. 5.4 - Velocity The graph shows the velocity, in feet per...Ch. 5.4 - Velocity The graph shows the velocity, in feet per...Ch. 5.4 - Particle Motion In Exercises 93-98, the velocity...Ch. 5.4 - Particle Motion In Exercises 93-98, the velocity...Ch. 5.4 - Particle Motion In Exercises 93-98, the velocity...Ch. 5.4 - Particle Motion In Exercises 93-98, the velocity...Ch. 5.4 - Particle Motion In Exercises 93-98, the velocity...Ch. 5.4 - Particle Motion In Exercises 93-98, the velocity...Ch. 5.4 - EXPLORING CONCEPTS Particle Motion Describe a...Ch. 5.4 - EXPLORING CONCEPTS Rate of Growth Let r'(t)...Ch. 5.4 - EXPLORING CONCEPTS Fundamental Theorem of Calculus...Ch. 5.4 - Modeling Data An experimental vehicle is tested on...Ch. 5.4 - Particle Motion A particle is moving along the...Ch. 5.4 - Particle Motion Repeat Exercise 103 for the...Ch. 5.4 - Error Analysis In Exercises 105-108, describe why...Ch. 5.4 - Error Analysis In Exercises 105-108, describe why...Ch. 5.4 - Error Analysis In Exercises 105-108, describe why...Ch. 5.4 - Error Analysis In Exercises 105-108, describe why...Ch. 5.4 - True or False? In Exercises 109 and 110, determine...Ch. 5.4 - True or False? In Exercises 109 and 110, 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. 5.4 - PUTNAM EXAM CHALLENGE For each continuous function...Ch. 5.5 - CONCEPT CHECK Constant Multiple Rule Explain how...Ch. 5.5 - CONCEPT CHECK Change of Variables In your own...Ch. 5.5 - CONCEPT CHECK The General Power Rule for...Ch. 5.5 - CONCEPT CHECK Analyzing the Integrand Without...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 - 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 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 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 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 29-32, find 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 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 - 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 65-72, find the...Ch. 5.5 - Change of Variables In Exercises 65-72, 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 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 - 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 - EXPLORING CONCEPTS Choosing an Integral You are...Ch. 5.5 - EXPLORING CONCEPTS Comparing Methods Find the...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 - 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 - The probability that a person will remember...Ch. 5.5 - The probability that ore samples taken from a...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 - 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 - CONCEPT CHECK LHopitals Rule Explain the benefit...Ch. 5.6 - CONCEPT CHECK Indeterminate Forms For each limit,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7- 14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7- 14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7- 14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7- 14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7- 14, evaluate the...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - EXPLORING CONCEPTS Finding Functions Find...Ch. 5.6 - EXPLORING CONCEPTS Finding Functions Find...Ch. 5.6 - EXPLORING CONCEPTS L' Hospitals Rule Determine...Ch. 5.6 - HOW DO YOU SEE IT? Use the graph of f to find each...Ch. 5.6 - Numerical Analysis Complete the table to show that...Ch. 5.6 - Numerical Analysis Complete the table to show that...Ch. 5.6 - Comparing Functions In Exercises 69-74, use L'...Ch. 5.6 - Comparing Functions In Exercises 69-74, use L'...Ch. 5.6 - Comparing Functions In Exercises 69-74, use L'...Ch. 5.6 - Comparing Functions In Exercises 69-74, use L'...Ch. 5.6 - Comparing Functions In Exercises 69-74, use L'...Ch. 5.6 - Comparing Functions In Exercises 69-74, use L'...Ch. 5.6 - Asymptotes and Relative Extrema In Exercises...Ch. 5.6 - Asymptotes and Relative Extrema In Exercises...Ch. 5.6 - Asymptotes and Relative Extreme In Exercises...Ch. 5.6 - Asymptotes and Relative Extreme In Exercises...Ch. 5.6 - Think About It In Exercises 79-82, L' Hospitals...Ch. 5.6 - Think About It In Exercises 79-82, L' Hospitals...Ch. 5.6 - Think About It In Exercises 79-82, L' Hospitals...Ch. 5.6 - Think About It In Exercises 79-82, L' Hospitals...Ch. 5.6 - Analytic and Graphical Analysis In Exercises 83...Ch. 5.6 - Analytic and Graphical Analysis In Exercises 83...Ch. 5.6 - Graphical Analysis In Exercises 85 and 86, graph...Ch. 5.6 - Graphical Analysis In Exercises 85 and 86, graph...Ch. 5.6 - Electric Circuit The diagram shows a simple...Ch. 5.6 - Velocity in a Resisting Medium The velocity of an...Ch. 5.6 - The Gamma Function The Gamma Function (n) is...Ch. 5.6 - Compound Interest The formula for the amount A in...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - Area Find the limit, as x approaches 0, of the...Ch. 5.6 - Finding a Limit In Section 2.3, a geometric...Ch. 5.6 - Continuous Function In Exercises 103 and 104, find...Ch. 5.6 - Continuous Function In Exercises 103 and 104, find...Ch. 5.6 - Finding Values Find the values of a and h such...Ch. 5.6 - Evaluating a Limit Use a graphing utility to graph...Ch. 5.6 - Finding a Derivative (a) Let f'(x) be continuous....Ch. 5.6 - Finding a Second Derivative Lei f"(x) be...Ch. 5.6 - Evaluating a Limit Consider the limit limx0+(xlnx)...Ch. 5.6 - Proof Prove that if f(x)0,limxaf(x)=0 and...Ch. 5.6 - Proof prove that if f(x)0,limxaf(x)=0 and...Ch. 5.6 - Think About It Use two different methods to find...Ch. 5.6 - Indeterminate Forms Show that the indeterminate...Ch. 5.6 - Calculus History In L' Hospital s 1696 calculus...Ch. 5.6 - Finding Limit consider the function h(x)=x+sinxx...Ch. 5.6 - Evaluating a Limit Let f(x)=x+xsinx and g(x)=x24....Ch. 5.6 - PUTNAM EXAM CHALLENGE Evaluate limx[1xax1a1]1/x...Ch. 5.7 - CONCEPT CHECK Log Rule for Integration Can you use...Ch. 5.7 - CONCEPT CHECK Long Division Explain when to use...Ch. 5.7 - CONCEPT CHECK Guidelines for Integration Describe...Ch. 5.7 - CONCEPT CHECK Trigonometric Functions Integrating...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 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 - Change of Variables In Exercises 29-32, find the...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 - 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 - 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 - 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 61-64, find F (...Ch. 5.7 - Finding a Derivative In Exercises 61-64, find F (...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 - 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 - Midpoint Rule In Exercises 77 and 78, use the...Ch. 5.7 - Midpoint Rule In Exercises 77 and 78, use the...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 - Napier's Inequality For 0xy, use the Mean Value...Ch. 5.7 - Think About It Is the function F(x)=x2x1tdt...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? Use the graph of f shown in the...Ch. 5.7 - True or False? In Exercises 97-100, determine...Ch. 5.7 - True or False? In Exercises 97-100, determine...Ch. 5.7 - True or False? In Exercises 97-100, determine...Ch. 5.7 - True or False? In Exercises 97-100, determine...Ch. 5.7 - Proof Prove Theorem 5.19.Ch. 5.7 - PUTNAM EXAM CHALLENGE Suppose that f is a function...Ch. 5.8 - CONCEPT CHECK Integration Rules Decide whether you...Ch. 5.8 - CONCEPT CHECK Completing the Square In your own...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Completing the Square In Exercises 35-42, Find or...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - EXPLORING CONCEPTS Comparing Antiderivatives In...Ch. 5.8 - EXPLORING CONCEPTS Comparing Antiderivatives In...Ch. 5.8 - Inverse Trigonometric Functions The antiderivative...Ch. 5.8 - HOW DO YOU SEE IT? 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 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? In Exercises 71 and 72, determine...Ch. 5.8 - True or False? In Exercises 71 and 72, 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 - CONCEPT CHECK Hyperbolic Functions Describe how...Ch. 5.9 - CONCEPT CHECK Domains of Hyperbolic Functions...Ch. 5.9 - CONCEPT CHECK Hyperbolic Identities Which...Ch. 5.9 - CONCEPT CHECK Derivatives of Inverse Hyperbolic...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 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 - 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 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 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 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 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 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 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 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 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 45-54,...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 55-60,...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 55-60,...Ch. 5.9 - EXPLORING CONCEPTS Using a Graph Explain...Ch. 5.9 - EXPLORING CONCEPTS Hyperbolic Functions Use the...Ch. 5.9 - Think About It Verify the results of Exercise 62...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 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 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 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 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 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 - 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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Approximating Area with the Midpoint Rule In...Ch. 5 - Evaluating a Definite Integral as a Limit In...Ch. 5 - Evaluating a Definite Integral as a Limit In...Ch. 5 - Evaluating a Definite Integral Using a Geometric...Ch. 5 - Evaluating a Definite Integral Using a Geometric...Ch. 5 - Using Properties of Definite Integrals Given...Ch. 5 - Using Properties of Definite Integrals Given...Ch. 5 - Evaluating a Definite Integral In Exercises 35-40,...Ch. 5 - Evaluating a Definite Integral In Exercises 35-40,...Ch. 5 - Evaluating a Definite Integral In Exercises 35-40,...Ch. 5 - Evaluating a Definite Integral In Exercises 35-40,...Ch. 5 - Evaluating a Definite Integral In Exercises 35-40,...Ch. 5 - Evaluating a Definite Integral In Exercises 35-40,...Ch. 5 - Finding the Area of a Region In Exercises 41-44,...Ch. 5 - Finding the Area of a Region In Exercises 41-44,...Ch. 5 - Finding the Area of a Region In Exercises 41-44,...Ch. 5 - Finding the Area of a Region In Exercises 41-44,...Ch. 5 - Using the Mean Value Theorem for Integrals In...Ch. 5 - Using the Mean Value Theorem for Integrals In...Ch. 5 - Finding the Average Value of a Function In...Ch. 5 - Finding the Average Value of a Function In...Ch. 5 - Using the Second Fundamental Theorem of Calculus...Ch. 5 - Using the Second Fundamental Theorem of Calculus...Ch. 5 - Using the Second Fundamental Theorem of Calculus...Ch. 5 - Using the Second Fundamental Theorem of Calculus...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - Finding an Indefinite Integral In Exercises 53-62,...Ch. 5 - 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