2.12.E: Exercises
- Page ID
- 157574
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.12 Exercises
In problems 1 – 10 find \(dy/dx\) by differentiating implicitly then find the value of \(dy/dx\) at the given point.
| 1. \(x^2 + y^2 = 100\), point (6, 8) | 2. \(x^2 + 5y^2 = 45\), point (5, 2) |
| 3. \(x^2 – 3xy + 7y = 5\), point (2,1) | 4. \(\sqrt{x} + \sqrt{y} = 5\), point (4,9) |
| 5. \(\frac{x^2}{9} + \frac{y^2}{16} = 1\), point (0,4) | 6. \(\frac{x^2}{9} + \frac{y^2}{16} = 1\), point (3,0) |
| 7. \(\ln(y) + 3x – 7 = 0\), point (2,\(e\)) | 8. \(x^2 – y^2 = 16\), point (5,3) |
| 9. \(x^2 – y^2 = 16\), point (5, –3) | 10. \(y^2 + 7x^3 – 3x = 8\), point (1,2) |
| 11. Find the slopes of the lines tangent to the graph in shown at the points (3,1), (3,3), and (4,2). |
| 12. Find the slopes of the lines tangent to the graph in shown where the graph crosses the \(y\)–axis. |
| 13. Find the slopes of the lines tangent to the graph in graph shown at the points ((5,0), (5,6), and (–4,3). |
| 14. Find the slopes of the lines tangent to the graph in the graph shown where the graph crosses the \(y\)–axis. |
In problems 15 – 16, find \(dy/dx\) using implicit differentiation and then find the slope of the line tangent to the graph of the equation at the given point.
| 15. \(y^3 – 5y = 5x^2 + 7\), point (1,3) | 16. \(y^2 – 5xy + x^2 + 21 = 0\), point (2,5) |
An expandable sphere is being filled with liquid at a constant rate from a tap (imagine a water balloon connected to a faucet). When the radius of the sphere is 3 inches, the radius is increasing at 2 inches per minute. How fast is the liquid coming out of the tap? \(( V = \frac{4}{3} \pi r^3 )\)
The 12 inch base of a right triangle is growing at 3 inches per hour, and the 16 inch height is shrinking at 3 inches per hour.
(a) Is the area increasing or decreasing?
(b) Is the perimeter increasing or decreasing?
(c) Is the hypotenuse increasing or decreasing?
One hour later the right triangle in Problem 2 is 15 inches long and 13 inches high, and the base and height are changing at the same rate as in Problem 18.
(a) Is the area increasing or decreasing now?
(b) Is the hypotenuse increasing or decreasing now?
(c) Is the perimeter increasing or decreasing now?
A young woman and her boyfriend plan to elope, but she must rescue him from his mother who has locked him in his room. The young woman has placed a 20 foot long ladder against his house and is knocking on his window when his mother begins pulling the bottom of the ladder away from the house at a rate of 3 feet per second. How fast is the top of the ladder (and the young couple) falling when the bottom of the ladder is
(a) 12 feet from the bottom of the wall?
(b) 16 feet from the bottom of the wall?
(c) 19 feet from the bottom of the wall?
The length of a 12 foot by 8 foot rectangle is increasing at a rate of 3 feet per second and the width is decreasing at 2 feet per second.
(a) How fast is the perimeter changing?
(b) How fast is the area changing?
An oil tanker in Puget Sound has sprung a leak, and a circular oil slick is forming. The oil slick is 4 inches thick everywhere, is 100 feet in diameter, and the diameter is increasing at 12 feet per hour. Your job, as the Coast Guard commander or the tanker's captain, is to determine how fast the oil is leaking from the tanker.