Skip to main content

Solutions 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.

Chapter 1 Limits, Continuity and Derivatives

Section 1.1 The notion of limit

Exercises 1.1.4 Exercises
1.1.4.1. Limits on a piecewise graph.
1.1.4.2. Estimating a limit numerically.
1.1.4.3. Limits for a piecewise formula.
1.1.4.4. Evaluating a limit algebraically.

Section 1.2 The derivative

Exercises 1.2.4 Exercises
1.2.4.1. The derivative function graphically.
1.2.4.2. Applying the limit definition of the derivative.
1.2.4.3. Sketching the derivative.
1.2.4.4. Comparing function and derivative values.
1.2.4.5. Limit definition of the derivative for a rational function.

Section 1.3 The second derivative

Exercises 1.3.5 Exercises
1.3.5.1. Comparing \(f, f', f''\) values.
1.3.5.2. Signs of \(f, f', f''\) values.
1.3.5.3. Acceleration from velocity.
1.3.5.4. Rates of change of stock values.
1.3.5.5. Interpreting a graph of \(f'\).

Section 1.4 Limits, Continuity, and Differentiability

Exercises 1.4.5 Exercises
1.4.5.1. Limit values of a piecewise graph.
1.4.5.2. Limit values of a piecewise formula.
1.4.5.3. Continuity and differentiability of a graph.
1.4.5.4. Continuity of a piecewise formula.

Section 1.5 The Tangent Line Approximation

Exercises 1.5.4 Exercises
1.5.4.1. Approximating \(\sqrt{x}\).
1.5.4.2. Local linearization of a graph.
1.5.4.3. Estimating with the local linearization.
1.5.4.4. Predicting behavior from the local linearization.

Chapter 2 Computing Derivatives

Section 2.1 Elementary derivative rules

Exercises 2.1.5 Exercises

Section 2.2 The sine and cosine functions

Exercises 2.2.3 Exercises

Section 2.3 The product and quotient rules

Exercises 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.

Section 2.4 Derivatives of other trigonometric functions

Exercises 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.

Section 2.5 The chain rule

Exercises 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.

Section 2.6 Derivatives of Inverse Functions

Exercises 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.

Section 2.7 Derivatives of Functions Given Implicitly

Exercises 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.

Chapter 3 Using Derivatives

Section 3.1 Using Derivatives to Evaluate Limits

Exercises 3.1.4 Exercises
3.1.4.1. L'Hôpital's Rule with graphs.
3.1.4.2. L'Hôpital's Rule to evaluate a limit.
3.1.4.3. Determining if L'Hôpital's Rule applies.
3.1.4.4. Using L'Hôpital's Rule multiple times.

Section 3.2 Using derivatives to identify extreme values

Exercises 3.2.4 Exercises
3.2.4.1. Finding critical points and inflection points.
3.2.4.2. Finding inflection points.
3.2.4.3. Matching graphs of \(f,f',f''\).

Section 3.3 Using derivatives to describe families of functions

Exercises 3.3.3 Exercises
3.3.3.1. Drug dosage with a parameter.
3.3.3.2. Using the graph of \(g'\).

Section 3.4 Applied Optimization

Exercises 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.

Section 3.5 Related Rates

Exercises 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.

Chapter 4 The Definite Integral

Section 4.1 Determining distance traveled from velocity

Exercises 4.1.5 Exercises
4.1.5.1. Estimating distance traveled from velocity data.
4.1.5.2. Distance from a linear veloity 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.

Section 4.2 Riemann Sums

Exercises 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.

Section 4.3 The Definite Integral

Exercises 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.

Section 4.4 The Fundamental Theorem of Calculus

Exercises 4.4.5 Exercises
4.4.5.1. Finding exact displacement.
4.4.5.2. Evaluating the definite integral of a rational function.

Chapter 5 Evaluating Integrals

Section 5.1 Constructing Accurate Graphs of Antiderivatives

Exercises 5.1.5 Exercises
5.1.5.1. Definite integral of a piecewise linear function.
5.1.5.2. A smooth function that starts out at 0.
5.1.5.3. A piecewise constant function.
5.1.5.4. Another piecewise linear function.

Section 5.2 The Second Fundamental Theorem of Calculus

Exercises 5.2.5 Exercises
5.2.5.1. A definite integral starting at 3.
5.2.5.2. Variable in the lower limit.
5.2.5.3. Approximating a function with derivative \(e^{-x^2/5}\).

Section 5.3 Integration by Substitution

Exercises 5.3.5 Exercises
5.3.5.1. Product involving 4th power of a polynomial.
5.3.5.2. Product involving \(\sin(x^6)\).
5.3.5.3. Fraction involving \(\ln^9\).
5.3.5.4. Fraction involving \(e^{5 x}\).
5.3.5.5. Fraction involving \(e^{5 \sqrt{y}}\).
5.3.5.6. Definite integral involving \(e^{-cos(q)}\).

Section 5.4 Integration by Parts

Exercises 5.4.7 Exercises
5.4.7.2. Product involving \(\cos(5 x)\).
5.4.7.3. Product involving \(e^{8 z}\).
5.4.7.4. Definite integral of \(t e^{-t}\).

Section 5.5 Other Options for Finding Algebraic Antiderivatives

Exercises 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.

Section 5.6 Numerical Integration

Exercises 5.6.6 Exercises
5.6.6.1. Various methods for \(e^x\) numerically.
5.6.6.2. Comparison of methods for increasing concave down function.

Chapter 6 Using Definite Integrals

Section 6.1 Using Definite Integrals to Find Area and Length

Exercises 6.1.5 Exercises
6.1.5.1. Area between two power functions.
6.1.5.2. Area between two trigonometric functions.
6.1.5.3. Area between two curves.
6.1.5.4. Arc length of a curve.

Section 6.2 Using Definite Integrals to Find Volume

Exercises 6.2.3 Exercises
6.2.3.1. Solid of revolution from one function about the \(x\)-axis.
6.2.3.2. Solid of revolution from one function about the \(y\)-axis.
6.2.3.3. Solid of revolution from two functions about the \(x\)-axis.
6.2.3.4. Solid of revolution from two functions about a horizontal line.
6.2.3.5. Solid of revolution from two functions about a different horizontal line.
6.2.3.6. Solid of revolution from two functions about a vertical line.

Section 6.3 Density, Mass, and Center of Mass

Exercises 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.

Section 6.4 Physics Applications: Work, Force, and Pressure

Exercises 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.

Section 6.5 Improper Integrals

Exercises 6.5.5 Exercises
6.5.5.1. An improper integral on a finite interval.
6.5.5.2. An improper integral on an infinite interval.
6.5.5.3. An improper integral involving a ratio of exponential functions.
6.5.5.4. A subtle improper integral.
6.5.5.5. An improper integral involving a ratio of trigonometric functions.

Chapter 7 Differential Equations

Section 7.1 An Introduction to Differential Equations

Exercises 7.1.5 Exercises
7.1.5.2. Finding constant to complete solution.

Section 7.2 Qualitative behavior of solutions to DEs

Exercises 7.2.4 Exercises
7.2.4.1. Graphing equilibrium solutions.
7.2.4.2. Sketching solution curves.
7.2.4.4. Describing equilibrium solutions.

Section 7.3 Euler's method

Exercises 7.3.4 Exercises
7.3.4.1. A few steps of Euler's method.
7.3.4.2. Using Euler's method for a solution of \(y'=4y\).
7.3.4.3. Using Euler's method with different time steps.

Section 7.4 Separable differential equations

Exercises 7.4.3 Exercises
7.4.3.1. Initial value problem for \(dy/dx=x^8 y\).
7.4.3.2. Initial value problem for \(dy/dt=0.9(y-300)\).
7.4.3.3. Initial value problem for \(dy/dt=y^2(8+t)\).
7.4.3.4. Initial value problem for \(du/dt=e^{6u+10t}\).
7.4.3.5. Initial value problem for \(dy/dx=170yx^{16}\).

Section 7.5 Modeling with differential equations

Exercises 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.

Section 7.6 Population Growth and the Logistic Equation

Exercises 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.

Chapter 8 Sequences and Series

Section 8.1 Sequences

Exercises 8.1.3 Exercises
8.1.3.2. Formula for a sequence, given first terms.
8.1.3.3. Divergent or convergent sequences.
8.1.3.4. Terms of a sequence from sampling a signal.

Section 8.2 Geometric Series

Exercises 8.2.3 Exercises
8.2.3.1. Seventh term of a geometric sequence.
8.2.3.2. A geometric series.
8.2.3.3. A series that is not geometric.
8.2.3.4. Two sums of geometric sequences.

Section 8.3 Series of Real Numbers

Exercises 8.3.7 Exercises
8.3.7.1. Convergence of a sequence and its series.
8.3.7.2. Two partial sums.
8.3.7.3. Convergence of a series and its sequence.
8.3.7.4. Convergence of an integral and a related series.

Section 8.4 Alternating Series

Exercises 8.4.6 Exercises
8.4.6.2. Estimating the sum of an alternating series.
8.4.6.3. Estimating the sum of a different alternating series.
8.4.6.4. Estimating the sum of one more alternating series.

Section 8.5 Taylor Polynomials and Taylor Series

Exercises 8.5.6 Exercises
8.5.6.1. Determining Taylor polynomials from a function formula.
8.5.6.2. Determining Taylor polynomials from given derivative values.
8.5.6.3. Finding the Taylor series for a given rational function.
8.5.6.4. Finding the Taylor series for a given trigonometric function.
8.5.6.5. Finding the Taylor series for a given logarithmic function.

Section 8.6 Power Series

Exercises 8.6.4 Exercises
8.6.4.1. Finding coefficients in a power series expansion of a rational function.
8.6.4.2. Finding coefficients in a power series expansion of a function involving \(\arctan(x)\).
login