3.2E: Exercises
- Page ID
- 10625
This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
For the following exercises, assume that \(f(x)\) and \(g(x)\) are both differentiable functions for all \(x\). Find the derivative of each of the functions \(h(x)\).
1) \(h(x)=4f(x)+\frac{g(x)}{7}\)
2) \(h(x)=x^3f(x)\)
3) \(h(x)=\frac{f(x)g(x)}{2}\)
4) \(h(x)=\frac{3f(x)}{g(x)+2}\)
- Answers to even numbered questions
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2. \(h′(x)=3x^2f(x)+x^3f′(x)\)
4. \(h′(x)=\frac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}\)
Exercise \(\PageIndex{2}\)
For the following exercises, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.
| \(x\) | 1 | 2 | 3 | 4 |
| \(f(x)\) | 3 | 5 | −2 | 0 |
| \(g(x)\) | 2 | 3 | −4 | 6 |
| \(f′(x)\) | −1 | 7 | 8 | −3 |
| \(g′(x)\) | 4 | 1 | 2 | 9 |
1) Find \(h′(1)\) if \(h(x)=x f(x)+4g(x)\).
2) Find \(h′(2)\) if \(h(x)=\frac{f(x)}{g(x)}\).
- Answer
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\(\frac{16}{9}\)
- Solution
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By quotient rule, \(h'(x)=\frac{g(x)f'(x)-f(x) g'(x)}{g^2(x)}\). Hence \(h'(2)=\frac{g(2)f'(2)-f(2) g'(2)}{g^2(2)}=\frac{(3)(7)-(5) (1)}{3^2}=\frac{16}{9}\).
3) Find \(h′(3)\) if \(h(x)=2x+f(x)g(x)\).
4) Find \(h′(4)\) if \(h(x)=\frac{1}{x}+\frac{g(x)}{f(x)}\).
- Answer
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Undefined
Exercise \(\PageIndex{3}\)
For the following exercises, use the following figure to find the indicated derivatives, if they exist.
1) Let \(h(x)=f(x)+g(x)\). Find
a. \(h′(1)\),
b. \(h′(3)\), and
c. \(h′(4)\).
2) Let \(h(x)=f(x)g(x).\) Find
a. \(h′(1),\)
b. \(h′(3)\), and
c. \(h′(4).\)
3) Let \(h(x)=\frac{f(x)}{g(x)}.\) Find
a. \(h′(1),\)
b. \(h′(3)\), and
c. \(h′(4).\)
- Solution to even numbered question:
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Let \(h(x)=f(x)g(x).\) Notice that from the graph,
a) \(f(1)=3,f(3)=1,\) and \( f(4)=2\)
b)\(g(1)=3,g(3)=2.5,\) and \( g(4)=2.5\)
c) The rate of change of \(f(x)\) (slope),\(f′(x)\), is \(-1\) when \(x=1\), is \(DNE \) when \(x=3\) and is \(1\) when \(x=4\).
d) Also, the rate of change of \(g(x)\) (slope) is \(1\) when \(x=1\), is \(0 \) when \(x=3\) and is \(0\) when \(x=4\).
Now we can solve the problem:
a. \(h′(1)=f′(1)g(1)+f(1)g′(1)=(-1)(1)+(3)(1)=2,\)
b. \(h′(3)=f′(3)g(3)+f(3)g′(3)=DNE\), and
c. \(h′(4)=f′(4)g(4)+f(4)g′(4)=(1)(2.5)+(2)(0)=2.5.\)
- Answers to even numbered questions
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2a. \(2\)
b. does not exist
c. \(2.5\)
Exercise \(\PageIndex{4}\)
For the following exercises,
a. evaluate \(f′(a)\), and
b. graph the function \(f(x)\) and the tangent line at \(x=a.\)
1) \(f(x)=2x^3+3x−x^2,a=2\)
2) \(f(x)=\frac{1}{x}−x^2,a=1\)
3) \(f(x)=x^2−x^{12}+3x+2,a=0\)
4) \(f(x)=\frac{1}{x}−x^{2/3},a=−1\)
- Answers to odd numbered questions
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1a. 23
b. \(y=23x−28\)
3a. 3
b. \(y=3x+2\)
Exercise \(\PageIndex{5}\)
Find the derivative of the following functions and simplify your answer:
1) \(f(x)=x^7+10\)
2) \(f(x)=5x^3−x+1\)
3) \(f(x)=4x^2−7x\)
4) \(f(x)=8x^4+9x^2−1\)
5) \(f(x)=x^4+2x\)
6) \(f(x)=3x(18x^4+\frac{13}{x+1})\)
7) \(f(x)=(x+2)(2x^2−3)\)
8) \(f(x)=x^2(\frac{2}{x^2}+\frac{5}{x^3})\)
9) \(f(x)=\frac{x^3+2x^2−4}{3}\)
10) \(f(x)=\frac{4x^3−2x+1}{x^2}\)
11) \(f(x)=\frac{x^2+4}{x^2−4}\)
12) \(f(x)=\frac{x+9}{x^2−7x+1}\)
13) \(\displaystyle{y(x)=\frac{3x^2+5}{2x^2+x-3}}\).
14) \(\displaystyle{y(x)=\frac{3x^2+5}{2x^2+x-3}}\)
- Answers to even numbered questions
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2. \(f′(x)=15x^2−1\)
4. \(f′(x)=32x^3+18x\)
6.\(f′(x)=270x^4+\frac{39}{(x+1)^2}\)
8. \(f′(x)=\frac{−5}{x^2}\)
10. \(f′(x)=\frac{4x^4+2x^2−2x}{x^4}\)
12. \(f′(x)=\frac{−x^2−18x+64}{(x^2−7x+1)^2}\)
Exercise \(\PageIndex{6}\)
For the following exercises, find the equation of the tangent line \(T(x)\) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.
1) \(y=3x^2+4x+1\) at \((0,1)\)
2) \(y=2\sqrt{x}+1\) at \((4,5)\)
3) \(y=\frac{2x}{x−1}\) at \((−1,1)\)
4) \(y=\frac{2}{x}−\frac{3}{x^2}\) at \((1,−1)\)
- Answers to even numbered questions
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2. \(T(x)=\frac{1}{2}x+3\),
4. \(T(x)=4x−5\)
Exercise \(\PageIndex{7}\)
1) Find the equation of the tangent line to the graph of \(f(x)=2x^3+4x^2−5x−3\) at \(x=−1.\)
2) Find the equation of the tangent line to the graph of \(f(x)=x^2+\frac{4}{x}−10\) at \(x=8\).
3) Find the equation of the tangent line to the graph of \(f(x)=(3x−x^2)(3−x−x^2)\) at \(x=1\).
4) Find the point on the graph of \(f(x)=x^3\) such that the tangent line at that point has an x intercept of 6.
5) Find the equation of the line passing through the point \(P(3,3)\) and tangent to the graph of \(f(x)=\frac{6}{x−1}\).
- Answers to odd numbered questions
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1. \(y=−7x−3\)
3. \(y=−5x+7\)
5. \(y=−\frac{3}{2}x+\frac{15}{2}\)
Exercise \(\PageIndex{8}\)
Determine all points on the graph of \(f(x)=x^3+x^2−x−1\) for which the slope of the tangent line is
a. horizontal
b. −1.
- Answer
- Under Construction
Exercise \(\PageIndex{9}\)
Find a quadratic polynomial such that \(f(1)=5,f′(1)=3\) and \(f''(1)=−6.\)
- Answer
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\(y=−3x^2+9x−1\)
Exercise \(\PageIndex{10}\)
A car driving along a freeway with traffic has traveled \(s(t)=t^3−6t^2+9t\) meters in \(t\) seconds.
a. Determine the time in seconds when the velocity of the car is 0.
b. Determine the acceleration of the car when the velocity is 0.
- Answer
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Under Construction
Exercise \(\PageIndex{11}\)
A herring swimming along a straight line has traveled \(s(t)=\frac{t^2}{t^2+2}\) feet in \(t\) seconds.
Determine the velocity of the herring when it has traveled 3 seconds.
- Answer
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\(\frac{12}{121}\) or 0.0992 ft/s
Exercise \(\PageIndex{12}\)
The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function \(P(t)=\frac{8t+3}{0.2t^2+1}\), where \(t\) is measured in years.
a. Determine the initial flounder population.
b. Determine \(P′(10)\) and briefly interpret the result.
- Answer
- Under Construction
Exercise \(\PageIndex{12}\)
The concentration of antibiotic in the bloodstream \(t\) hours after being injected is given by the function \(C(t)=\frac{2t^2+t}{t^3+50}\), where \(C\) is measured in milligrams per liter of blood.
a. Find the rate of change of \(C(t).\)
b. Determine the rate of change for \(t=8,12,24\),and \(36\).
c. Briefly describe what seems to be occurring as the number of hours increases.
- Answer
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\(a. \frac{−2t^4−2t^3+200t+50}{(t^3+50)^2}\) \(b. −0.02395\) mg/L-hr, −0.01344 mg/L-hr, −0.003566 mg/L-hr, −0.001579 mg/L-hr c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.mplate active on the page.
Exercise \(\PageIndex{13}\)
A book publisher has a cost function given by \(C(x)=\frac{x^3+2x+3}{x^2}\), where x is the number of copies of a book in thousands and C is the cost per book, measured in dollars. Evaluate \(C′(2)\) and explain its meaning.
- Answer
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Under Construction
Exercise \(\PageIndex{14}\)
According to Newton’s law of universal gravitation, the force \(F\) between two bodies of constant mass \(m_1\) and \(m_2\) is given by the formula \(F=\frac{Gm_1m_2}{d^2}\), where \(G\) is the gravitational constant and \(d\) is the distance between the bodies.
a. Suppose that \(G,m_1,\) and \(m_2\) are constants. Find the rate of change of force \(F\) with respect to distance \(d\).
b. Find the rate of change of force \(F\) with gravitational constant \(G=6.67×10^{−11} Nm^2/kg^2\), on two bodies 10 meters apart, each with a mass of 1000 kilograms.
- Answer
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a. \(F'(d)=\frac{−2Gm_1m_2}{d_3}\) '
b. \( −1.33×10^{−7} N/m\)
Contributors and Attributions
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.