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Calculus Volume 1

Review Exercises

Calculus Volume 1Review Exercises
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? Justify the answer with a proof or a counterexample.

367.

Every function has a derivative.

368.

A continuous function has a continuous derivative.

369.

A continuous function has a derivative.

370.

If a function is differentiable, it is continuous.

Use the limit definition of the derivative to exactly evaluate the derivative.

371.

f ( x ) = x + 4 f ( x ) = x + 4

372.

f ( x ) = 3 x f ( x ) = 3 x

Find the derivatives of the following functions.

373.

f ( x ) = 3 x 3 − 4 x 2 f ( x ) = 3 x 3 − 4 x 2

374.

f ( x ) = ( 4 − x 2 ) 3 f ( x ) = ( 4 − x 2 ) 3

375.

f ( x ) = e sin x f ( x ) = e sin x

376.

f ( x ) = ln ( x + 2 ) f ( x ) = ln ( x + 2 )

377.

f ( x ) = x 2 cos x + x tan ( x ) f ( x ) = x 2 cos x + x tan ( x )

378.

f ( x ) = 3 x 2 + 2 f ( x ) = 3 x 2 + 2

379.

f ( x ) = x 4 sin −1 ( x ) f ( x ) = x 4 sin −1 ( x )

380.

x 2 y = ( y + 2 ) + x y sin ( x ) x 2 y = ( y + 2 ) + x y sin ( x )

Find the following derivatives of various orders.

381.

First derivative of y=xln(x)cosxy=xln(x)cosx

382.

Third derivative of y=(3x+2)2y=(3x+2)2

383.

Second derivative of y=4x+x2sin(x)y=4x+x2sin(x)

Find the equation of the tangent line to the following equations at the specified point.

384.

y=cos−1(x)+xy=cos−1(x)+x at x=0x=0

385.

y=x+ex−1xy=x+ex−1x at x=1x=1

Draw the derivative for the following graphs.

386.
The function begins at (−3, 0.5) and decreases to a local minimum at (−2.3, −2). Then the function increases through (−1.5, 0) and slows its increase through (0, 2). It then slowly increases to a local maximum at (2.3, 6) before decreasing to (3, 3).
387.
The function decreases linearly from (−1, 4) to the origin, at which point it increases as x2, passing through (1, 1) and (2, 4).

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by w(t)=1.9+2.9cos(π6t),w(t)=1.9+2.9cos(π6t), where t is measured in hours after midnight, and the height is measured in feet.

388.

Find and graph the derivative. What is the physical meaning?

389.

Find w′(3).w′(3). What is the physical meaning of this value?

The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.

Hours after Midnight, August 26 Wind Speed (mph)
1 45
5 75
11 100
29 115
49 145
58 175
73 155
81 125
85 95
107 35
Table 3.9 Wind Speeds of Hurricane Katrina Source: http://news.nationalgeographic.com/news/2005/09/0914_050914_katrina_timeline.html.
390.

Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?

391.

Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

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