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Chapter 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 5.1.3 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 Sums

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