Active Calculus

Appendix B Answers to Activities

1 Understanding the Derivative
1.1 How do we measure velocity?
1.1.1 Position and average velocity

1.1.2 Instantaneous Velocity

1.2 The notion of limit
1.2.1 The Notion of Limit

1.2.2 Instantaneous Velocity

1.3 The derivative of a function at a point
1.3.1 The Derivative of a Function at a Point

1.4 The derivative function
1.4.1 How the derivative is itself a function

1.5 Interpreting, estimating, and using the derivative
1.5.2 Toward more accurate derivative estimates

1.6 The second derivative
1.6.3 Concavity

1.7 Limits, Continuity, and Differentiability
1.7.1 Having a limit at a point

1.7.2 Being continuous at a point

1.7.3 Being differentiable at a point

1.8 The Tangent Line Approximation
1.8.2 The local linearization

2 Computing Derivatives
2.1 Elementary derivative rules
2.1.2 Constant, Power, and Exponential Functions

2.1.3 Constant Multiples and Sums of Functions

2.2 The sine and cosine functions
2.2.1 The sine and cosine functions

2.3 The product and quotient rules
2.3.1 The product rule

2.3.2 The quotient rule

2.3.3 Combining rules

2.4 Derivatives of other trigonometric functions
2.4.1 Derivatives of the cotangent, secant, and cosecant functions

2.5 The chain rule
2.5.1 The chain rule

2.5.2 Using multiple rules simultaneously

2.6 Derivatives of Inverse Functions
2.6.2 The derivative of the natural logarithm function

2.6.3 Inverse trigonometric functions and their derivatives

2.7 Derivatives of Functions Given Implicitly
2.7.1 Implicit Differentiation

2.8 Using Derivatives to Evaluate Limits
2.8.1 Using derivatives to evaluate indeterminate limits of the form \(\frac{0}{0}\text{.}\)

2.8.2 Limits involving \(\infty\)

3 Using Derivatives
3.1 Using derivatives to identify extreme values
3.1.1 Critical numbers and the first derivative test

3.1.2 The second derivative test

3.2 Using derivatives to describe families of functions
3.2.1 Describing families of functions in terms of parameters

3.3 Global Optimization
3.3.1 Global Optimization

3.3.2 Moving toward applications

3.4 Applied Optimization
3.4.1 More applied optimization problems

3.5 Related Rates
3.5.1 Related Rates Problems

4 The Definite Integral
4.1 Determining distance traveled from velocity
4.1.1 Area under the graph of the velocity function

4.1.2 Two approaches: area and antidifferentiation

4.1.3 When velocity is negative

4.2 Riemann Sums
4.2.1 Sigma Notation

4.2.2 Riemann Sums

4.2.3 When the function is sometimes negative

4.3 The Definite Integral
4.3.1 The definition of the definite integral

4.3.2 Some properties of the definite integral

4.3.3 How the definite integral is connected to a function's average value

4.4 The Fundamental Theorem of Calculus
4.4.1 The Fundamental Theorem of Calculus

4.4.2 Basic antiderivatives

4.4.3 The total change theorem

5 Evaluating Integrals
5.1 Constructing Accurate Graphs of Antiderivatives
5.1.1 Constructing the graph of an antiderivative

5.1.2 Multiple antiderivatives of a single function

5.1.3 Functions defined by integrals

5.2 The Second Fundamental Theorem of Calculus
5.2.1 The Second Fundamental Theorem of Calculus

5.2.2 Understanding Integral Functions

5.2.3 Differentiating an Integral Function

5.3 Integration by Substitution
5.3.1 Reversing the Chain Rule: First Steps

5.3.2 Reversing the Chain Rule: \(u\)-substitution

5.3.3 Evaluating Definite Integrals via \(u\)-substitution

5.4 Integration by Parts
5.4.1 Reversing the Product Rule: Integration by Parts

5.4.2 Some Subtleties with Integration by Parts

5.4.3 Using Integration by Parts Multiple Times

5.5 Other Options for Finding Algebraic Antiderivatives
5.5.1 The Method of Partial Fractions

5.5.2 Using an Integral Table

5.6 Numerical Integration
5.6.1 The Trapezoid Rule

5.6.3 Simpson's Rule

5.6.4 Overall observations regarding \(L_n\text{,}\) \(R_n\text{,}\) \(T_n\text{,}\) \(M_n\text{,}\) and \(S_{2n}\text{.}\)

6 Using Definite Integrals
6.1 Using Definite Integrals to Find Area and Length
6.1.1 The Area Between Two Curves

6.1.2 Finding Area with Horizontal Slices

6.1.3 Finding the length of a curve

6.2 Using Definite Integrals to Find Volume
6.2.1 The Volume of a Solid of Revolution

6.2.2 Revolving about the \(y\)-axis

6.2.3 Revolving about horizontal and vertical lines other than the coordinate axes

6.3 Density, Mass, and Center of Mass
6.3.1 Density

6.3.2 Weighted Averages

6.3.3 Center of Mass

6.4 Physics Applications: Work, Force, and Pressure
6.4.1 Work

6.4.2 Work: Pumping Liquid from a Tank

6.4.3 Force due to Hydrostatic Pressure

6.5 Improper Integrals
6.5.1 Improper Integrals Involving Unbounded Intervals

6.5.2 Convergence and Divergence

6.5.3 Improper Integrals Involving Unbounded Integrands

7 Differential Equations
7.1 An Introduction to Differential Equations
7.1.1 What is a differential equation?

7.1.2 Differential equations in the world around us

7.1.3 Solving a differential equation

7.2 Qualitative behavior of solutions to DEs
7.2.1 Slope fields

7.2.2 Equilibrium solutions and stability

7.3 Euler's method
7.3.1 Euler's Method

7.4 Separable differential equations
7.4.1 Solving separable differential equations

7.5 Modeling with differential equations
7.5.1 Developing a differential equation

7.6 Population Growth and the Logistic Equation
7.6.1 The earth's population

7.6.2 Solving the logistic differential equation

8 Taylor Polynomials and Taylor Series
8.1 Approximating \(f(x) = e^x\)
8.1.1 Finding a quadratic approximation

8.1.2 Over and over again

8.1.3 As the degree of the approximation increases

8.2 Taylor Polynomials
8.2.1 Taylor polynomials

8.2.2 Taylor polynomial approximations centered at an arbitrary value \(a\)

8.3 Geometric Sums
8.3.1 Finite Geometric Series

8.3.2 Infinite Geometric Series

8.3.3 How geometric series naturally connect to Taylor polynomials

8.4 Taylor Series
8.4.1 Taylor series and the Ratio Test

8.4.2 Taylor series of several important functions

8.5 Finding and Using Taylor Series
8.5.1 Using substitution and algebra to find new Taylor series expressions

8.5.2 Differentiating and integrating Taylor series

8.5.3 Alternating series of real numbers