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Calculus Volume 1

Review Exercises

Calculus Volume 1Review Exercises
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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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

Review Exercises

True or False. In the following exercises, justify your answer with a proof or a counterexample.

208.

A function has to be continuous at x=ax=a if the limx→af(x)limx→af(x) exists.

209.

You can use the quotient rule to evaluate limx→0sinxx.limx→0sinxx.

210.

If there is a vertical asymptote at x=ax=a for the function f(x),f(x), then f is undefined at the point x=a.x=a.

211.

If limx→af(x)limx→af(x) does not exist, then f is undefined at the point x=a.x=a.

212.

Using the graph, find each limit or explain why the limit does not exist.

  1. limx→−1f(x)limx→−1f(x)
  2. limx→1f(x)limx→1f(x)
  3. limx→0+f(x)limx→0+f(x)
  4. limx→2f(x)limx→2f(x)
A graph of a piecewise function with several segments. The first is a decreasing concave up curve existing for x < -1. It ends at an open circle at (-1, 1). The second is an increasing linear function starting at (-1, -2) and ending at (0,-1). The third is an increasing concave down curve existing from an open circle at (0,0) to an open circle at (1,1). The fourth is a closed circle at (1,-1). The fifth is a line with no slope existing for x > 1, starting at the open circle at (1,1).

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

213.

lim x → 2 2 x 2 − 3 x − 2 x − 2 lim x → 2 2 x 2 − 3 x − 2 x − 2

214.

lim x → 0 3 x 2 − 2 x + 4 lim x → 0 3 x 2 − 2 x + 4

215.

lim x → 3 x 3 − 2 x 2 − 1 3 x − 2 lim x → 3 x 3 − 2 x 2 − 1 3 x − 2

216.

lim x → π / 2 cot x cos x lim x → π / 2 cot x cos x

217.

lim x → −5 x 2 + 25 x + 5 lim x → −5 x 2 + 25 x + 5

218.

lim x → 2 3 x 2 − 2 x − 8 x 2 − 4 lim x → 2 3 x 2 − 2 x − 8 x 2 − 4

219.

lim x → 1 x 2 − 1 x 3 − 1 lim x → 1 x 2 − 1 x 3 − 1

220.

lim x → 1 x 2 − 1 x − 1 lim x → 1 x 2 − 1 x − 1

221.

lim x → 4 4 − x x − 2 lim x → 4 4 − x x − 2

222.

lim x → 4 1 x − 2 lim x → 4 1 x − 2

In the following exercises, use the squeeze theorem to prove the limit.

223.

lim x → 0 x 2 cos ( 2 π x ) = 0 lim x → 0 x 2 cos ( 2 π x ) = 0

224.

lim x → 0 x 3 sin ( π x ) = 0 lim x → 0 x 3 sin ( π x ) = 0

225.

Determine the domain such that the function f(x)=x−2+xexf(x)=x−2+xex is continuous over its domain.

In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.

226.

f ( x ) = { x 2 + 1 , x > c 2 x , x ≤ c f ( x ) = { x 2 + 1 , x > c 2 x , x ≤ c

227.

f ( x ) = { x + 1 , x > − 1 x 2 + c , x ≤ − 1 f ( x ) = { x + 1 , x > − 1 x 2 + c , x ≤ − 1

In the following exercises, use the precise definition of limit to prove the limit.

228.

lim x → 1 ( 8 x + 16 ) = 24 lim x → 1 ( 8 x + 16 ) = 24

229.

lim x → 0 x 3 = 0 lim x → 0 x 3 = 0

230.

A ball is thrown into the air and the vertical position is given by x(t)=−4.9t2+25t+5.x(t)=−4.9t2+25t+5. Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

231.

A particle moving along a line has a displacement according to the function x(t)=t2−2t+4,x(t)=t2−2t+4, where x is measured in meters and t is measured in seconds. Find the average velocity over the time period t=[0,2].t=[0,2].

232.

From the previous exercises, estimate the instantaneous velocity at t=2t=2 by checking the average velocity within t=0.01sec.t=0.01sec.

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