3.3E: Exercises for Section 3.3
- Page ID
- 96024
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In exercises 1 - 4, find \(f'(x)\) for each function.
1) \(f(x)=(x+2)(2x^2−3)\)
2) \(f(x)=\dfrac{4x^3−2x+1}{x^2}\)
- Answer
- \(f'(x) = \dfrac{4x^4+2x^2−2x}{x^4}\)
3) \(f(x)=\dfrac{x^2+4}{x^2−4}\)
4) \(f(x)=\dfrac{x+9}{x^2−7x+1}\)
- Answer
- \(f'(x) = \dfrac{−x^2−18x+64}{(x^2−7x+1)^2}\)
In exercises 5 - 8, assume that \(f(x)\) and \(g(x)\) are both differentiable functions for all \(x\). Find the derivative of each of the functions \(h(x)\).
5) \(h(x)=4f(x)+\dfrac{g(x)}{7}\)
6) \(h(x)=x^3f(x)\)
- Answer
- \(h'(x)=3x^2f(x)+x^3f′(x)\)
7) \(h(x)=\dfrac{f(x)g(x)}{2}\)
8) \(h(x)=\dfrac{3f(x)}{g(x)+2}\)
- Answer
- \(h'(x)=\dfrac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}\)
For exercises 9 - 12, 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 |
9) Find \(h′(1)\) if \(h(x)=x f(x)+4g(x)\).
10) Find \(h′(2)\) if \(h(x)=\dfrac{f(x)}{g(x)}\).
- Answer
- \(h'(2) =\frac{16}{9}\)
11) Find \(h′(3)\) if \(h(x)=2x+f(x)g(x)\).
12) Find \(h′(4)\) if \(h(x)=\dfrac{1}{x}+\dfrac{g(x)}{f(x)}\).
- Answer
- \(h'(4)\) is undefined.
In exercises 13 - 15, use the following figure to find the indicated derivatives, if they exist.
13) Let \(h(x)=f(x)+g(x)\). Find
a) \(h′(1)\),
b) \(h′(3)\), and
c) \(h′(4)\).
14) Let \(h(x)=f(x)g(x).\) Find
a) \(h′(1),\)
b) \(h′(3)\), and
c) \(h′(4).\)
- Answer
- a. \(h'(1) = 2\),
b. \(h'(3)\) does not exist,
c. \(h'(4) = 2.5\)
15) Let \(h(x)=\dfrac{f(x)}{g(x)}.\) Find
a) \(h′(1),\)
b) \(h′(3)\), and
c) \(h′(4).\)
16) Find the equation of the tangent line to the graph of \(f(x)=(3x−x^2)(3−x−x^2)\) at \(x=1\).
- Answer
- \(y=−5x+7\)
17) [T] A herring swimming along a straight line has traveled \(s(t)=\dfrac{t^2}{t^2+2}\) feet in \(t\)
seconds. Determine the velocity of the herring when it has traveled 3 seconds.
- Answer
- \(\frac{12}{121}\) or 0.0992 ft/s
18) [T] The concentration of antibiotic in the bloodstream \(t\) hours after being injected is given by the function \(C(t)=\dfrac{2t^2+t}{t^3+50}\), 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
- a. \(\dfrac{−2t^4−2t^3+200t+50}{(t^3+50)^2}\)
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.
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.