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Calculus Volume 1

Chapter Review Exercises

Calculus Volume 1Chapter Review Exercises
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  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Key Terms
    13. Key Equations
    14. Key Concepts
    15. Chapter Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index

True or False. Justify your answer with a proof or a counterexample. Assume all functions ff and gg are continuous over their domains.

439.

If f(x)>0,f(x)>0f(x)>0,f(x)>0 for all x,x, then the right-hand rule underestimates the integral abf(x).abf(x). Use a graph to justify your answer.

440.

abf(x)2dx=abf(x)dxabf(x)dxabf(x)2dx=abf(x)dxabf(x)dx

441.

If f(x)g(x)f(x)g(x) for all x[a,b],x[a,b], then abf(x)abg(x).abf(x)abg(x).

442.

All continuous functions have an antiderivative.

Evaluate the Riemann sums L4andR4L4andR4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

443.

y=3x22x+1y=3x22x+1 over [−1,1][−1,1]

444.

y=ln(x2+1)y=ln(x2+1) over [0,e][0,e]

445.

y=x2sinxy=x2sinx over [0,π][0,π]

446.

y=x+1xy=x+1x over [1,4][1,4]

Evaluate the following integrals.

447.

−11(x32x2+4x)dx−11(x32x2+4x)dx

448.

043t1+6t2dt043t1+6t2dt

449.

π/3π/22sec(2θ)tan(2θ)dθπ/3π/22sec(2θ)tan(2θ)dθ

450.

0π/4ecos2xsinxcosxdx0π/4ecos2xsinxcosxdx

Find the antiderivative.

451.

dx(x+4)3dx(x+4)3

452.

xln(x2)dxxln(x2)dx

453.

4x21x6dx4x21x6dx

454.

e2x1+e4xdxe2x1+e4xdx

Find the derivative.

455.

ddt0tsinx1+x2dxddt0tsinx1+x2dx

456.

ddx1x34t2dtddx1x34t2dt

457.

ddx1ln(x)(4t+et)dtddx1ln(x)(4t+et)dt

458.

ddx0cosxet2dtddx0cosxet2dt

The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year 5-Year Change ($)
1980 0
1985 −5,468,750
1990 755,495
1995 −73,005
2000 −29,768
2005 −918
2010 −177
459.

If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

460.

The average cost per gigabyte of RAM can be approximated by the function C(t)=8,500,000(0.65)t,C(t)=8,500,000(0.65)t, where tt is measured in years since 1980, and CC is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

461.

Find the average cost of 1GB RAM for 2005 to 2010.

462.

The velocity of a bullet from a rifle can be approximated by v(t)=6400t26505t+2686,v(t)=6400t26505t+2686, where tt is seconds after the shot and vv is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: 0t0.5.0t0.5. What is the total distance the bullet travels in 0.5 sec?

463.

What is the average velocity of the bullet for the first half-second?

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