2.5.E: Exercises
- Page ID
- 157567
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.5 Exercises
Use the values in the table to fill in the rest of the table.
| \(x\) | \(f(x)\) | \(f'(x)\) | \(g(x)\) | \(g'(x)\) | \(\frac{d}{dx} (f(x) \cdot g(x))\) | \(\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)\) | \(\frac{d}{dx}\left(\frac{g(x)}{f(x)}\right)\) |
| 0 | 3 | -2 | -4 | 3 | |||
| 1 | 2 | -1 | 1 | 0 | |||
| 2 | 4 | 2 | 3 | 1 |
2. Use the information in the graph to plot the values of the functions \(f + g\), \(f \cdot g\) and \(f/g\) and their derivatives at \(x =\) 1, 2 and 3.
3. Use the information in the graph to plot the values of the functions \(2f\), \(f – g\) and \(g/f\) and their derivatives at \(x =\) 1, 2 and 3.
Calculate \(\frac{d}{dx} ((x-5)(3x+7))\) by (a) using the product rule and (b) expanding the product and then differentiating. Verify that both methods give the same result.
If the product of \(f\) and \(g\) is a constant \(( f(x) \cdot g(x) = k\) for all \(x\)), then how are \(\frac{\frac{d}{dx}(f(x))}{f(x)}\) and \(\frac{\frac{d}{dx}(g(x))}{g(x)}\) related?
If the quotient of \(f\) and \(g\) is a constant ( \(\frac{f(x)}{g(x)} = k\) for all \(x\)), then how are \(g \cdot f'\) and \(f \cdot g '\) related?
In problems 7 – 8, (a) calculate \(f '(1)\) and (b) determine when \(f '(x) = 0\)
| 7. \(f(x) = \frac{7x}{x^2+4}\) | 8. \(f(x) = \frac{3x^2}{2x-3}\) |
Determine \(\frac{d}{dx}(x^2 + 1)(7x - 3)\) and \(\frac{d}{dt}(\frac{3t-2}{5t+1})\).
Find (a) \(\frac{d}{dx}(x^3e^x)\) and (b) \(\frac{d}{dx}(e^x)^3\).
Find (a) \(\frac{d}{dt} (te^t)\), (b) \(d(e^x)^5\)
A manufacturer has determined that an employee with d days of production experience will be able to produce approximately \(P(d) = 3 + 15( 1 – e^{–0.2d} )\) items per day. Graph \(P(d)\).
(a) Approximately how many items will a beginning employee be able to produce each day?
(b) How many items will an experienced employee be able to produce each day?
(c) What is the marginal production rate of an employee with 5 days of experience? (What are the units of your answer, and what does this answer mean?