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Active Calculus

Appendix C Answers to Selected Exercises

This appendix contains answers to all non-WeBWorK exercises in the text. For WeBWorK exercises, please use the HTML version of the text for access to answers and solutions.

1 Understanding the Derivative
1.1 How do we measure velocity?
1.1.4 Exercises Matching a distance graph to velocity.

1.2 The notion of limit
1.2.4 Exercises Estimating a limit numerically. Evaluating a limit algebraically.

1.3 The derivative of a function at a point
1.3.3 Exercises Finding an exact derivative value algebraically. Estimating a derivative from the limit definition.

1.4 The derivative function
1.4.3 Exercises Sketching the derivative.

1.5 Interpreting, estimating, and using the derivative
1.5.4 Exercises

1.6 The second derivative
1.6.5 Exercises

1.7 Limits, Continuity, and Differentiability
1.7.5 Exercises Continuity of a piecewise formula.

1.8 The Tangent Line Approximation
1.8.4 Exercises Estimating with the local linearization.

2 Computing Derivatives
2.1 Elementary derivative rules
2.1.5 Exercises Derivative of a power function. Derivative of a rational function. Derivative of a root function. Derivative of a quadratic. Derivative of a sum of power functions. Simplifying a product before differentiating. Simplifying a quotient before differentiating. Finding a tangent line equation. Determining where \(f'(x) = 0\).

2.2 The sine and cosine functions
2.2.3 Exercises

2.3 The product and quotient rules
2.3.5 Exercises Derivative of a basic product. Derivative of a product. Derivative of a quotient of linear functions. Derivative of a rational function. Derivative of a product of trigonometric functions. Derivative of a product of power and trigonmetric functions. Derivative of a sum that involves a product.

2.4 Derivatives of other trigonometric functions
2.4.3 Exercises A sum and product involving \(\tan(x)\).

2.5 The chain rule
2.5.5 Exercises Mixing rules: chain, product, sum. Mixing rules: chain and product. Using the chain rule repeatedly. Derivative involving arbitrary constants \(a\) and \(b\). A product involving a composite function.

2.6 Derivatives of Inverse Functions
2.6.6 Exercises Composite function involving logarithms and polynomials. Composite function involving trigonometric functions and logarithms. Product involving \(\arcsin(w)\). Derivative involving \(\arctan(x)\). Composite function involving an inverse trigonometric function. Mixing rules: product, chain, and inverse trig. Mixing rules: product and inverse trig.

2.7 Derivatives of Functions Given Implicitly
2.7.3 Exercises Implicit differentiaion in a polynomial equation. Implicit differentiation in an equation with logarithms. Implicit differentiation in an equation with inverse trigonometric functions. Slope of the tangent line to an implicit curve. Equation of the tangent line to an implicit curve.

2.8 Using Derivatives to Evaluate Limits
2.8.4 Exercises L’Hôpital’s Rule to evaluate a limit. Using L’Hôpital’s Rule multiple times.

3 Using Derivatives
3.1 Using derivatives to identify extreme values
3.1.4 Exercises Finding inflection points.

3.2 Using derivatives to describe families of functions
3.2.3 Exercises

3.3 Global Optimization
3.3.4 Exercises

3.4 Applied Optimization
3.4.3 Exercises Minimizing the cost of a container. Maximizing area contained by a fence.

3.5 Related Rates
3.5.3 Exercises Height of a conical pile of gravel. Movement of a shadow. A leaking conical tank.

4 The Definite Integral
4.1 Determining distance traveled from velocity
4.1.5 Exercises Distance from a linear velocity function. Change in position from a linear velocity function. Change in position from a quadratic velocity function.

4.2 Riemann Sums
4.2.5 Exercises

4.3 The Definite Integral
4.3.5 Exercises Finding the average value of a linear function. Estimating a definite integral and average value from a graph.

4.4 The Fundamental Theorem of Calculus
4.4.5 Exercises Finding exact displacement. Evaluating the definite integral of a rational function. Evaluating the definite integral of a linear function. Evaluating the definite integral of a quadratic function. Simplifying an integrand before integrating. Evaluating the definite integral of a trigonometric function.

5 Evaluating Integrals
5.1 Constructing Accurate Graphs of Antiderivatives
5.1.5 Exercises

5.2 The Second Fundamental Theorem of Calculus
5.2.5 Exercises

5.3 Integration by Substitution
5.3.5 Exercises

5.4 Integration by Parts
5.4.7 Exercises

5.5 Other Options for Finding Algebraic Antiderivatives
5.5.5 Exercises Partial fractions: constant over product. Partial fractions: linear over quadratic.

5.6 Numerical Integration
5.6.6 Exercises

6 Using Definite Integrals
6.1 Using Definite Integrals to Find Area and Length
6.1.5 Exercises

6.2 Using Definite Integrals to Find Volume
6.2.5 Exercises

6.3 Density, Mass, and Center of Mass
6.3.5 Exercises

6.4 Physics Applications: Work, Force, and Pressure
6.4.5 Exercises Work to empty a conical tank. Work to empty a cylindrical tank. Work to empty a rectangular pool.

6.5 Improper Integrals
6.5.5 Exercises

7 Differential Equations
7.1 An Introduction to Differential Equations
7.1.5 Exercises

7.2 Qualitative behavior of solutions to DEs
7.2.4 Exercises

7.3 Euler’s method
7.3.4 Exercises

7.4 Separable differential equations
7.4.3 Exercises

7.5 Modeling with differential equations
7.5.3 Exercises

7.6 Population Growth and the Logistic Equation
7.6.4 Exercises

8 Taylor Polynomials and Taylor Series
8.1 Approximating \(f(x) = e^x\)
8.1.5 Exercises

8.2 Taylor Polynomials
8.2.4 Exercises

8.3 Geometric Sums
8.3.5 Exercises

8.4 Taylor Series
8.4.4 Exercises

8.5 Finding and Using Taylor Series
8.5.5 Exercises

9 Supplementary material
9.1 Review of Prerequsites


9.2 Integrating Rational Functions
9.2.2 Exercises

9.3 Integration with Trigonometric Functions
9.3.2 Exercises

9.4 Parametric Curves


9.5 Calculus in Polar Coordinates