2.4.E: Exercises
- Page ID
- 157566
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.4 Exercises
Fill in the values in the table for \(\frac{d}{dx} (3(f(x))\), \(\frac{d}{dx}(2f(x)+g(x))\), and \(\frac{d}{dx}(3(g(x)-f(x))\).
| \(x\) | \(f(x)\) | \(f'(x)\) | \(g(x)\) | \(g'(x)\) | \(\frac{d}{dx} (3(f(x))\) | \(\frac{d}{dx}(2f(x)+g(x))\) | \(\frac{d}{dx}(3(g(x)-f(x))\) |
| 0 | 3 | -2 | -4 | 3 | |||
| 1 | 2 | -1 | 1 | 0 | |||
| 2 | 4 | 2 | 3 | 1 |
Find
(a) \(D( x^{12} )\)
(b) \(\frac{d}{dx} (\sqrt[7]{x})\)
(c) \(D(\frac{1}{x^3})\)
(d) \(\frac{d x^e}{dx}\)
Find
(a) \(D( x^{9} )\)
(b) \(\frac{d x^{2/3}}{dx}\)
(c) \(D(\frac{1}{x^4})\)
(d) \(D(x^{\pi})\)
In problems 4 – 8, (a) calculate \(f '(1)\) and (b) determine when \(f '(x) = 0\).
| 4. \(f(x) = x^2 – 5x + 13\) |
| 5. \(f(x) = 5x^2 – 40x + 73\) |
| 6. \(f(x) = x^3 + 9x^2 + 6\) |
| 7. \(f(x) = x^3 + 3x^2 + 3x – 1\) |
| 8. \(f(x) = x^3 + 2x^2 + 2x – 1\) |
Where do \(f(x) = x^2 – 10x + 3\) and \(g(x) = x^3 – 12x\) have horizontal tangent lines?
It takes \(T(x) = x^2\) hours to weave \(x\) small rugs. What is the marginal production time to weave a rug? (Be sure to include the units with your answer.)
It costs \(C(x) = \sqrt{x}\) dollars to produce x golf balls. What is the marginal production cost to make a golf ball? What is the marginal production cost when \(x = 25\)? when \(x= 100\)? (Include units.)
An arrow shot straight up from ground level with an initial velocity of 128 feet per second will be at height \(h(x) = –16x^2 + 128x\) feet at \(x\) seconds.
(a) Determine the velocity of the arrow when \(x =\) 0, 1 and 2 seconds.
(b) What is the velocity of the arrow, \(v(x)\), at any time \(x\)?
(c) At what time \(x\) will the velocity of the arrow be 0?
(d) What is the greatest height the arrow reaches?
(e) How long will the arrow be aloft?
(f) Use the answer for the velocity in part (b) to determine the acceleration, \(a(x) = v '(x)\), at any time \(x\).
If an arrow is shot straight up from ground level on the moon with an initial velocity of 128 feet per second, its height will be \(h(x) = –2.65x^2 + 128x\) feet at \(x\) seconds. Do parts (a) – (e) of the previous exercise using this new equation for \(h\).
\(f(x) = x^3 + A x^2 + B x + C\) with constants \(A\), \(B\) and \(C\). Can you find conditions on the constants \(A\), \(B\) and \(C\) which will guarantee that the graph of \(y = f(x)\) has two distinct "vertices"? (Here a "vertex" means a place where the curve changes from increasing to decreasing or from decreasing to increasing.)