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Solutions B Answers to Activities

This appendix contains answers to all activities in the text. Answers for preview activities are not included.

Chapter 1 Understanding the Derivative

Section 1.1 How do we measure velocity?

Subsection 1.1.1 Position and average velocity
Subsection 1.1.2 Instantaneous Velocity

Section 1.2 The notion of limit

Subsection 1.2.1 The Notion of Limit
Subsection 1.2.2 Instantaneous Velocity

Section 1.3 The derivative of a function at a point

Subsection 1.3.1 The Derivative of a Function at a Point

Section 1.4 The derivative function

Subsection 1.4.1 How the derivative is itself a function

Section 1.5 Interpreting, estimating, and using the derivative

Subsection 1.5.2 Toward more accurate derivative estimates

Section 1.6 The second derivative

Subsection 1.6.3 Concavity

Section 1.7 Limits, Continuity, and Differentiability

Subsection 1.7.1 Having a limit at a point
Subsection 1.7.2 Being continuous at a point
Subsection 1.7.3 Being differentiable at a point

Section 1.8 The Tangent Line Approximation

Subsection 1.8.2 The local linearization

Chapter 2 Computing Derivatives

Section 2.1 Elementary derivative rules

Subsection 2.1.2 Constant, Power, and Exponential Functions
Subsection 2.1.3 Constant Multiples and Sums of Functions

Section 2.2 The sine and cosine functions

Subsection 2.2.1 The sine and cosine functions

Section 2.3 The product and quotient rules

Subsection 2.3.1 The product rule
Subsection 2.3.2 The quotient rule
Subsection 2.3.3 Combining rules

Section 2.4 Derivatives of other trigonometric functions

Subsection 2.4.1 Derivatives of the cotangent, secant, and cosecant functions

Section 2.5 The chain rule

Subsection 2.5.1 The chain rule
Subsection 2.5.2 Using multiple rules simultaneously

Section 2.6 Derivatives of Inverse Functions

Subsection 2.6.2 The derivative of the natural logarithm function
Subsection 2.6.3 Inverse trigonometric functions and their derivatives

Section 2.7 Derivatives of Functions Given Implicitly

Subsection 2.7.1 Implicit Differentiation

Section 2.8 Using Derivatives to Evaluate Limits

Subsection 2.8.1 Using derivatives to evaluate indeterminate limits of the form \(\frac{0}{0}\text{.}\)
Subsection 2.8.2 Limits involving \(\infty\)

Chapter 3 Using Derivatives

Section 3.1 Using derivatives to identify extreme values

Subsection 3.1.1 Critical numbers and the first derivative test
Subsection 3.1.2 The second derivative test

Section 3.2 Using derivatives to describe families of functions

Subsection 3.2.1 Describing families of functions in terms of parameters

Section 3.3 Global Optimization

Subsection 3.3.1 Global Optimization
Subsection 3.3.2 Moving toward applications

Section 3.4 Applied Optimization

Subsection 3.4.1 More applied optimization problems

Section 3.5 Related Rates

Subsection 3.5.1 Related Rates Problems

Chapter 4 The Definite Integral

Section 4.1 Determining distance traveled from velocity

Subsection 4.1.1 Area under the graph of the velocity function
Subsection 4.1.2 Two approaches: area and antidifferentiation
Subsection 4.1.3 When velocity is negative

Section 4.2 Riemann Sums

Subsection 4.2.1 Sigma Notation
Subsection 4.2.2 Riemann Sums
Subsection 4.2.3 When the function is sometimes negative

Section 4.3 The Definite Integral

Subsection 4.3.1 The definition of the definite integral
Subsection 4.3.2 Some properties of the definite integral
Subsection 4.3.3 How the definite integral is connected to a function's average value

Section 4.4 The Fundamental Theorem of Calculus

Subsection 4.4.1 The Fundamental Theorem of Calculus
Subsection 4.4.2 Basic antiderivatives
Subsection 4.4.3 The total change theorem

Chapter 5 Evaluating Integrals

Section 5.1 Constructing Accurate Graphs of Antiderivatives

Subsection 5.1.1 Constructing the graph of an antiderivative
Subsection 5.1.2 Multiple antiderivatives of a single function
Subsection 5.1.3 Functions defined by integrals

Section 5.2 The Second Fundamental Theorem of Calculus

Subsection 5.2.1 The Second Fundamental Theorem of Calculus
Subsection 5.2.2 Understanding Integral Functions
Subsection 5.2.3 Differentiating an Integral Function

Section 5.3 Integration by Substitution

Subsection 5.3.1 Reversing the Chain Rule: First Steps
Subsection 5.3.2 Reversing the Chain Rule: \(u\)-substitution
Subsection 5.3.3 Evaluating Definite Integrals via \(u\)-substitution

Section 5.4 Integration by Parts

Subsection 5.4.1 Reversing the Product Rule: Integration by Parts
Subsection 5.4.2 Some Subtleties with Integration by Parts
Subsection 5.4.3 Using Integration by Parts Multiple Times

Section 5.5 Other Options for Finding Algebraic Antiderivatives

Subsection 5.5.1 The Method of Partial Fractions
Subsection 5.5.2 Using an Integral Table

Section 5.6 Numerical Integration

Subsection 5.6.1 The Trapezoid Rule
Subsection 5.6.3 Simpson's Rule
Subsection 5.6.4 Overall observations regarding \(L_n\text{,}\) \(R_n\text{,}\) \(T_n\text{,}\) \(M_n\text{,}\) and \(S_{2n}\text{.}\)

Chapter 6 Using Definite Integrals

Section 6.1 Using Definite Integrals to Find Area and Length

Subsection 6.1.1 The Area Between Two Curves
Subsection 6.1.2 Finding Area with Horizontal Slices
Subsection 6.1.3 Finding the length of a curve

Section 6.2 Using Definite Integrals to Find Volume

Subsection 6.2.1 The Volume of a Solid of Revolution
Subsection 6.2.2 Revolving about the \(y\)-axis
Subsection 6.2.3 Revolving about horizontal and vertical lines other than the coordinate axes

Section 6.3 Density, Mass, and Center of Mass

Subsection 6.3.1 Density
Subsection 6.3.2 Weighted Averages
Subsection 6.3.3 Center of Mass

Section 6.4 Physics Applications: Work, Force, and Pressure

Subsection 6.4.1 Work
Subsection 6.4.2 Work: Pumping Liquid from a Tank
Subsection 6.4.3 Force due to Hydrostatic Pressure

Section 6.5 Improper Integrals

Subsection 6.5.1 Improper Integrals Involving Unbounded Intervals
Subsection 6.5.2 Convergence and Divergence
Subsection 6.5.3 Improper Integrals Involving Unbounded Integrands

Chapter 7 Differential Equations

Section 7.1 An Introduction to Differential Equations

Subsection 7.1.1 What is a differential equation?
Subsection 7.1.2 Differential equations in the world around us
Subsection 7.1.3 Solving a differential equation

Section 7.2 Qualitative behavior of solutions to DEs

Subsection 7.2.1 Slope fields
Subsection 7.2.2 Equilibrium solutions and stability

Section 7.3 Euler's method

Subsection 7.3.1 Euler's Method

Section 7.4 Separable differential equations

Subsection 7.4.1 Solving separable differential equations

Section 7.5 Modeling with differential equations

Subsection 7.5.1 Developing a differential equation

Section 7.6 Population Growth and the Logistic Equation

Subsection 7.6.1 The earth's population
Subsection 7.6.2 Solving the logistic differential equation

Chapter 8 Sequences and Series

Section 8.1 Sequences

Subsection 8.1.1 Sequences

Section 8.2 Geometric Series

Subsection 8.2.1 Geometric Series

Section 8.3 Series of Real Numbers

Subsection 8.3.1 Infinite Series
Subsection 8.3.2 The Divergence Test
Subsection 8.3.3 The Integral Test
Subsection 8.3.4 The Limit Comparison Test
Subsection 8.3.5 The Ratio Test

Section 8.4 Alternating Series

Subsection 8.4.1 The Alternating Series Test
Subsection 8.4.2 Estimating Alternating Sums
Subsection 8.4.3 Absolute and Conditional Convergence
Subsection 8.4.4 Summary of Tests for Convergence of Series

Section 8.5 Taylor Polynomials and Taylor Series

Subsection 8.5.1 Taylor Polynomials
Subsection 8.5.2 Taylor Series
Subsection 8.5.3 The Interval of Convergence of a Taylor Series
Subsection 8.5.4 Error Approximations for Taylor Polynomials

Section 8.6 Power Series

Subsection 8.6.1 Power Series
Subsection 8.6.2 Manipulating Power Series