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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
1.1.4.1. Average velocity from position.
1.1.4.2. Rate of calorie consumption.
1.1.4.3. Average rate of change - quadratic function.
1.1.4.4. Comparing average rate of change of two functions.
1.1.4.5. Matching a distance graph to velocity.
1.1.4.6. 1.1.4.7. 1.1.4.8.
1.2 The notion of limit 1.2.4 Exercises
1.2.4.1. Limits on a piecewise graph.
1.2.4.2. Estimating a limit numerically.
1.2.4.3. Limits for a piecewise formula.
1.2.4.4. Evaluating a limit algebraically.
1.2.4.5. 1.2.4.6. 1.2.4.7. 1.2.4.8.
1.3 The derivative of a function at a point 1.3.3 Exercises
1.3.3.1. Estimating derivative values graphically.
1.3.3.2. Tangent line to a curve.
1.3.3.3. Interpreting values and slopes from a graph.
1.3.3.4. Finding an exact derivative value algebraically.
1.3.3.5. Estimating a derivative from the limit definition.
1.3.3.6. 1.3.3.7. 1.3.3.8. 1.3.3.9.
1.4 The derivative function 1.4.3 Exercises
1.4.3.1. The derivative function graphically.
1.4.3.2. Applying the limit definition of the derivative.
1.4.3.3. Sketching the derivative.
1.4.3.4. Comparing function and derivative values.
1.4.3.5. Limit definition of the derivative for a rational function.
1.4.3.6. 1.4.3.7. 1.4.3.8. 1.4.3.9.
1.5 Interpreting, estimating, and using the derivative 1.5.4 Exercises
1.5.4.1. A cooling cup of coffee.
1.5.4.2. A cost function.
1.5.4.3. Weight as a function of calories.
1.5.4.4. Displacement and velocity.
1.5.4.5. 1.5.4.6. 1.5.4.7. 1.5.4.8.
1.6 The second derivative 1.6.5 Exercises
1.6.5.1. Comparing \(f, f', f''\) values.
1.6.5.2. Signs of \(f, f', f''\) values.
1.6.5.3. Acceleration from velocity.
1.6.5.4. Rates of change of stock values.
1.6.5.5. Interpreting a graph of \(f'\) .
1.6.5.6. 1.6.5.7. 1.6.5.8. 1.6.5.9.
1.7 Limits, Continuity, and Differentiability 1.7.5 Exercises
1.7.5.1. Limit values of a piecewise graph.
1.7.5.2. Limit values of a piecewise formula.
1.7.5.3. Continuity and differentiability of a graph.
1.7.5.4. Continuity of a piecewise formula.
1.7.5.5. 1.7.5.6. 1.7.5.7. 1.7.5.8.
1.8 The Tangent Line Approximation 1.8.4 Exercises
1.8.4.1. Approximating \(\sqrt{x}\) .
1.8.4.2. Local linearization of a graph.
1.8.4.3. Estimating with the local linearization.
1.8.4.4. Predicting behavior from the local linearization.
1.8.4.5. 1.8.4.6. 1.8.4.7. 1.8.4.8.
2 Computing Derivatives 2.1 Elementary derivative rules 2.1.5 Exercises
2.1.5.1. Derivative of a power function.
2.1.5.2. Derivative of a rational function.
2.1.5.3. Derivative of a root function.
2.1.5.4. Derivative of a quadratic.
2.1.5.5. Derivative of a sum of power functions.
2.1.5.6. Simplifying a product before differentiating.
2.1.5.7. Simplifying a quotient before differentiating.
2.1.5.8. Finding a tangent line equation.
2.1.5.9. Determining where \(f'(x) = 0\) .
2.1.5.10. 2.1.5.11. 2.1.5.12. 2.1.5.13.
2.2 The sine and cosine functions 2.2.3 Exercises
2.2.3.1. 2.2.3.2. 2.2.3.3.
2.3 The product and quotient rules 2.3.5 Exercises
2.3.5.1. Derivative of a basic product.
2.3.5.2. Derivative of a product.
2.3.5.3. Derivative of a quotient of linear functions.
2.3.5.4. Derivative of a rational function.
2.3.5.5. Derivative of a product of trigonometric functions.
2.3.5.6. Derivative of a product of power and trigonmetric functions.
2.3.5.7. Derivative of a sum that involves a product.
2.3.5.8. Product and quotient rules with graphs.
2.3.5.9. Product and quotient rules with given function values.
2.3.5.10. 2.3.5.11. 2.3.5.12. 2.3.5.13. 2.3.5.14.
2.4 Derivatives of other trigonometric functions 2.4.3 Exercises
2.4.3.1. A sum and product involving \(\tan(x)\) .
2.4.3.2. A quotient involving \(\tan(t)\) .
2.4.3.3. A quotient of trigonometric functions.
2.4.3.4. A quotient that involves a product.
2.4.3.5. Finding a tangent line equation.
2.4.3.6. 2.4.3.7. 2.4.3.8.
2.5 The chain rule 2.5.5 Exercises
2.5.5.1. Mixing rules: chain, product, sum.
2.5.5.2. Mixing rules: chain and product.
2.5.5.3. Using the chain rule repeatedly.
2.5.5.4. Derivative involving arbitrary constants \(a\) and \(b\) .
2.5.5.5. Chain rule with graphs.
2.5.5.6. Chain rule with function values.
2.5.5.7. A product involving a composite function.
2.5.5.8. 2.5.5.9. 2.5.5.10. 2.5.5.11.
2.6 Derivatives of Inverse Functions 2.6.6 Exercises
2.6.6.1. Composite function involving logarithms and polynomials.
2.6.6.2. Composite function involving trigonometric functions and logarithms.
2.6.6.3. Product involving \(\arcsin(w)\) .
2.6.6.4. Derivative involving \(\arctan(x)\) .
2.6.6.5. Composite function from a graph.
2.6.6.6. Composite function involving an inverse trigonometric function.
2.6.6.7. Mixing rules: product, chain, and inverse trig.
2.6.6.8. Mixing rules: product and inverse trig.
2.6.6.9. 2.6.6.10. 2.6.6.11. 2.6.6.12.
2.7 Derivatives of Functions Given Implicitly 2.7.3 Exercises
2.7.3.1. Implicit differentiaion in a polynomial equation.
2.7.3.2. Implicit differentiation in an equation with logarithms.
2.7.3.3. Implicit differentiation in an equation with inverse trigonometric functions.
2.7.3.4. Slope of the tangent line to an implicit curve.
2.7.3.5. Equation of the tangent line to an implicit curve.
2.7.3.6. 2.7.3.7. 2.7.3.8.
2.8 Using Derivatives to Evaluate Limits 2.8.4 Exercises
2.8.4.1. L’Hôpital’s Rule with graphs.
2.8.4.2. L’Hôpital’s Rule to evaluate a limit.
2.8.4.3. Determining if L’Hôpital’s Rule applies.
2.8.4.4. Using L’Hôpital’s Rule multiple times.
2.8.4.5. 2.8.4.6. 2.8.4.7. 2.8.4.8.
3 Using Derivatives 3.1 Using derivatives to identify extreme values 3.1.4 Exercises
3.1.4.1. Finding critical points and inflection points.
3.1.4.2. Finding inflection points.
3.1.4.3. Matching graphs of \(f,f',f''\) .
3.1.4.4. 3.1.4.5. 3.1.4.6. 3.1.4.7.
3.2 Using derivatives to describe families of functions 3.2.3 Exercises
3.2.3.1. Drug dosage with a parameter.
3.2.3.2. Using the graph of \(g'\) .
3.2.3.3. 3.2.3.4. 3.2.3.5.
3.3 Global Optimization 3.3.4 Exercises
3.3.4.1. 3.3.4.2. 3.3.4.3. 3.3.4.4.
3.4 Applied Optimization 3.4.3 Exercises
3.4.3.1. Maximizing the volume of a box.
3.4.3.2. Minimizing the cost of a container.
3.4.3.3. Maximizing area contained by a fence.
3.4.3.4. Minimizing the area of a poster.
3.4.3.5. Maximizing the area of a rectangle.
3.4.3.6. 3.4.3.7. 3.4.3.8. 3.4.3.9.
3.5 Related Rates 3.5.3 Exercises
3.5.3.1. Height of a conical pile of gravel.
3.5.3.2. Movement of a shadow.
3.5.3.3. A leaking conical tank.
3.5.3.4. 3.5.3.5. 3.5.3.6. 3.5.3.7.
4 The Definite Integral 4.1 Determining distance traveled from velocity 4.1.5 Exercises
4.1.5.1. Estimating distance traveled from velocity data.
4.1.5.2. Distance from a linear velocity function.
4.1.5.3. Change in position from a linear velocity function.
4.1.5.5. Finding average acceleration from velocity data.
4.1.5.6. Change in position from a quadratic velocity function.
4.1.5.7. 4.1.5.8. 4.1.5.9. 4.1.5.10.
4.2 Riemann Sums 4.2.5 Exercises
4.2.5.1. Evaluating Riemann sums for a quadratic function.
4.2.5.2. Estimating distance traveled with a Riemann sum from data.
4.2.5.3. Writing basic Riemann sums.
4.2.5.4. 4.2.5.5. 4.2.5.6. 4.2.5.7.
4.3 The Definite Integral 4.3.5 Exercises
4.3.5.1. Evaluating definite integrals from graphical information.
4.3.5.2. Estimating definite integrals from a graph.
4.3.5.3. Finding the average value of a linear function.
4.3.5.4. Finding the average value of a function given graphically.
4.3.5.5. Estimating a definite integral and average value from a graph.
4.3.5.6. Using rules to combine known integral values.
4.3.5.7. 4.3.5.8. 4.3.5.9. 4.3.5.10.
4.4 The Fundamental Theorem of Calculus 4.4.5 Exercises
4.4.5.1. Finding exact displacement.
4.4.5.2. Evaluating the definite integral of a rational function.
4.4.5.3. Evaluating the definite integral of a linear function.
4.4.5.4. Evaluating the definite integral of a quadratic function.
4.4.5.5. Simplifying an integrand before integrating.
4.4.5.6. Evaluating the definite integral of a trigonometric function.
4.4.5.7. 4.4.5.8. 4.4.5.9. 4.4.5.10. 4.4.5.11.
5 Evaluating Integrals 5.1 Constructing Accurate Graphs of Antiderivatives 5.1.5 Exercises
5.1.5.1. 5.1.5.2. 5.1.5.3. 5.1.5.4. 5.1.5.5. 5.1.5.6. 5.1.5.7.
5.2 The Second Fundamental Theorem of Calculus 5.2.5 Exercises
5.2.5.1. 5.2.5.2. 5.2.5.3. 5.2.5.4. 5.2.5.5. 5.2.5.6. 5.2.5.7.
5.3 Integration by Substitution 5.3.5 Exercises
5.3.5.1. 5.3.5.2. 5.3.5.3. 5.3.5.4. 5.3.5.5. 5.3.5.6. 5.3.5.7. 5.3.5.8. 5.3.5.9. 5.3.5.10. 5.3.5.11. 5.3.5.12. 5.3.5.13. 5.3.5.14.
5.4 Integration by Parts 5.4.7 Exercises
5.4.7.1. 5.4.7.2. 5.4.7.3. 5.4.7.4. 5.4.7.5. 5.4.7.6. 5.4.7.7. 5.4.7.8. 5.4.7.9. 5.4.7.10. 5.4.7.11. 5.4.7.13. 5.4.7.14. 5.4.7.15.
5.5 Other Options for Finding Algebraic Antiderivatives 5.5.5 Exercises
5.5.5.1. Partial fractions: linear over difference of squares.
5.5.5.2. Partial fractions: constant over product.
5.5.5.3. Partial fractions: linear over quadratic.
5.5.5.4. Partial fractions: cubic over 4th degree.
5.5.5.5. Partial fractions: quadratic over factored cubic.
5.5.5.6. 5.5.5.7. 5.5.5.8.
5.6 Numerical Integration 5.6.6 Exercises
5.6.6.1. 5.6.6.2. 5.6.6.3. 5.6.6.4. 5.6.6.5. 5.6.6.6. 5.6.6.7.
6 Using Definite Integrals 6.1 Using Definite Integrals to Find Area and Length 6.1.5 Exercises
6.1.5.1. 6.1.5.2. 6.1.5.3. 6.1.5.4. 6.1.5.5. 6.1.5.6. 6.1.5.7. 6.1.5.8. 6.1.5.9. 6.1.5.10. 6.1.5.11. 6.1.5.12.
6.2 Using Definite Integrals to Find Volume 6.2.5 Exercises
6.2.5.1. 6.2.5.2. 6.2.5.3. 6.2.5.4. 6.2.5.5. 6.2.5.7. 6.2.5.8. 6.2.5.9. 6.2.5.10. 6.2.5.11.
6.3 Density, Mass, and Center of Mass 6.3.5 Exercises
6.3.5.1. Center of mass for a linear density function.
6.3.5.2. Center of mass for a nonlinear density function.
6.3.5.3. Interpreting the density of cars on a road.
6.3.5.4. Center of mass in a point-mass system.
6.3.5.5. 6.3.5.6. 6.3.5.7.
6.4 Physics Applications: Work, Force, and Pressure 6.4.5 Exercises
6.4.5.1. Work to empty a conical tank.
6.4.5.2. Work to empty a cylindrical tank.
6.4.5.3. Work to empty a rectangular pool.
6.4.5.4. Work to empty a cylindrical tank to differing heights.
6.4.5.5. Force due to hydrostatic pressure.
6.4.5.6. 6.4.5.7.
6.5 Improper Integrals 6.5.5 Exercises
6.5.5.1. 6.5.5.2. 6.5.5.3. 6.5.5.4. 6.5.5.5. 6.5.5.6. 6.5.5.7. 6.5.5.8. 6.5.5.9. 6.5.5.10. 6.5.5.11. 6.5.5.12.
7 Differential Equations 7.1 An Introduction to Differential Equations 7.1.5 Exercises
7.1.5.3. 7.1.5.4. 7.1.5.6. 7.1.5.7. 7.1.5.8. 7.1.5.9.
7.2 Qualitative behavior of solutions to DEs 7.2.4 Exercises
7.2.4.1. 7.2.4.3. 7.2.4.4. 7.2.4.5. 7.2.4.6. 7.2.4.7. 7.2.4.8. 7.2.4.9.
7.3 Euler’s method 7.3.4 Exercises
7.3.4.1. 7.3.4.2. 7.3.4.3. 7.3.4.4. 7.3.4.5. 7.3.4.6. 7.3.4.7. 7.3.4.8.
7.4 Separable differential equations 7.4.3 Exercises
7.4.3.1. 7.4.3.2. 7.4.3.3. 7.4.3.4. 7.4.3.5. 7.4.3.6. 7.4.3.7. 7.4.3.8. 7.4.3.9. 7.4.3.10. 7.4.3.11.
7.5 Modeling with differential equations 7.5.3 Exercises
7.5.3.1. Mixing problem.
7.5.3.2. Mixing problem.
7.5.3.3. Population growth problem.
7.5.3.4. Radioactive decay problem.
7.5.3.5. Investment problem.
7.5.3.6. 7.5.3.7. 7.5.3.8. 7.5.3.9.
7.6 Population Growth and the Logistic Equation 7.6.4 Exercises
7.6.4.1. Analyzing a logistic equation.
7.6.4.2. Analyzing a logistic model.
7.6.4.3. Finding a logistic function for an infection model.
7.6.4.4. Analyzing a population growth model.
7.6.4.5. 7.6.4.6. 7.6.4.7.
8 Taylor Polynomials and Taylor Series 8.1 Approximating \(f(x) = e^x\) 8.1.5 Exercises
8.1.5.1. 8.1.5.2. 8.1.5.3. 8.1.5.4.
8.2 Taylor Polynomials 8.2.4 Exercises
8.2.4.1. 8.2.4.2. 8.2.4.3. 8.2.4.4. 8.2.4.5. 8.2.4.6. 8.2.4.8. 8.2.4.9. 8.2.4.10.
8.3 Geometric Sums 8.3.5 Exercises
8.3.5.1. 8.3.5.2. 8.3.5.3. 8.3.5.4. 8.3.5.5. 8.3.5.6. 8.3.5.7. 8.3.5.8. 8.3.5.9.
8.4 Taylor Series 8.4.4 Exercises
8.4.4.1. 8.4.4.2. 8.4.4.3. 8.4.4.4. 8.4.4.5. 8.4.4.6. 8.4.4.7. 8.4.4.8. 8.4.4.9.
8.5 Finding and Using Taylor Series 8.5.5 Exercises
8.5.5.1. 8.5.5.2. 8.5.5.2.a Part 1. 8.5.5.2.b Part 2. 8.5.5.2.c Part 3. 8.5.5.3. 8.5.5.4. 8.5.5.5. 8.5.5.6. 8.5.5.7. 8.5.5.8. 8.5.5.9. 8.5.5.10.
9 Supplementary material 9.1 Review of Prerequsites
Exercises 9.1.1. 9.1.2. 9.1.3. 9.1.4. 9.1.5. 9.1.6. 9.1.7. 9.1.8. 9.1.9. 9.1.10.
9.2 Integrating Rational Functions 9.2.2 Exercises
9.2.2.5. 9.2.2.6. 9.2.2.8.
9.3 Integration with Trigonometric Functions 9.3.2 Exercises
9.3.2.1. 9.3.2.2. 9.3.2.3. 9.3.2.4. 9.3.2.5. 9.3.2.6. 9.3.2.7. 9.3.2.8. 9.3.2.9.
9.4 Parametric Curves
Exercises 9.4.3. 9.4.4. 9.4.5. 9.4.6. 9.4.7. 9.4.8. 9.4.9. 9.4.10.
9.5 Calculus in Polar Coordinates
Exercises 9.5.1. 9.5.3. 9.5.4. 9.5.5. 9.5.6. 9.5.7. 9.5.8. 9.5.9. 9.5.10. 9.5.11.
