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$f\left(x\right)={2}^{4x}+4{x}^{2}$
${2}^{4x+2}\xb7\text{ln}\phantom{\rule{0.1em}{0ex}}2+8x$
$f\left(x\right)={3}^{\text{sin}\phantom{\rule{0.1em}{0ex}}3\text{x}}$
$f\left(x\right)={x}^{\pi}\xb7{\pi}^{x}$
$\pi {x}^{\pi -1}\xb7{\pi}^{x}+{x}^{\pi}\xb7{\pi}^{x}\text{ln}\phantom{\rule{0.1em}{0ex}}\pi $
$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(4{x}^{3}+x\right)$
$f\left(x\right)=\text{ln}\sqrt{5x-7}$
$\frac{5}{2\left(5x-7\right)}$
$f\left(x\right)={x}^{2}\text{ln}\phantom{\rule{0.1em}{0ex}}9x$
$f\left(x\right)=\text{log}\phantom{\rule{0.1em}{0ex}}\left(\text{sec}\phantom{\rule{0.1em}{0ex}}x\right)$
$\frac{\text{tan}\phantom{\rule{0.1em}{0ex}}x}{\text{ln}\phantom{\rule{0.1em}{0ex}}10}$
$f\left(x\right)={\text{log}}_{7}{\left(6{x}^{4}+3\right)}^{5}$
$f\left(x\right)={2}^{x}\xb7{\text{log}}_{3}{7}^{{x}^{2}-4}$
${2}^{x}\xb7\text{ln}\phantom{\rule{0.1em}{0ex}}2\xb7{\text{log}}_{3}{7}^{{x}^{2}-4}+{2}^{x}\xb7\frac{2x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}7}{\text{ln}\phantom{\rule{0.1em}{0ex}}3}$
For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$
$y={x}^{\sqrt{x}}$
$y={\left(\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right)}^{4x}$
${\left(\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right)}^{4x}\left[4\xb7\text{ln}\phantom{\rule{0.1em}{0ex}}\left(\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right)+8x\xb7\text{cot}\phantom{\rule{0.1em}{0ex}}2x\right]$
$y={\left(\text{ln}\phantom{\rule{0.1em}{0ex}}x\right)}^{\text{ln}\phantom{\rule{0.1em}{0ex}}x}$
$y={x}^{{\text{log}}_{2}x}$
${x}^{{\text{log}}_{2}x}\xb7\frac{2\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}x}{x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}2}$
$y={\left({x}^{2}-1\right)}^{\text{ln}\phantom{\rule{0.1em}{0ex}}x}$
$y={x}^{\text{cot}\phantom{\rule{0.1em}{0ex}}x}$
${x}^{\text{cot}\phantom{\rule{0.1em}{0ex}}x}\xb7\left[\text{\u2212}{\text{csc}}^{2}x\xb7\text{ln}\phantom{\rule{0.1em}{0ex}}x+\frac{\text{cot}\phantom{\rule{0.1em}{0ex}}x}{x}\right]$
$y=\frac{x+11}{\sqrt[3]{{x}^{2}-4}}$
$y={x}^{\mathrm{-1}\text{/}2}{\left({x}^{2}+3\right)}^{2\text{/}3}{\left(3x-4\right)}^{4}$
${x}^{\mathrm{-1}\text{/}2}{\left({x}^{2}+3\right)}^{2\text{/}3}{\left(3x-4\right)}^{4}\xb7\left[\frac{\mathrm{-1}}{2x}+\frac{4x}{3\left({x}^{2}+3\right)}+\frac{12}{3x-4}\right]$
[T] Find an equation of the tangent line to the graph of $f\left(x\right)=4x{e}^{\left({x}^{2}-1\right)}$ at the point where
$x=\mathrm{-1}.$ Graph both the function and the tangent line.
[T] Find the equation of the line that is normal to the graph of $f\left(x\right)=x\xb7{5}^{x}$ at the point where $x=1.$ Graph both the function and the normal line.
$y=\frac{\mathrm{-1}}{5+5\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}5}x+\left(5+\frac{1}{5+5\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}5}\right)$
[T] Find the equation of the tangent line to the graph of ${x}^{3}-x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}y+{y}^{3}=2x+5$ at the point where $x=2.$ ( Hint : Use implicit differentiation to find $\frac{dy}{dx}.)$ Graph both the curve and the tangent line.
Consider the function $y={x}^{1\text{/}x}$ for $x>0.$
a. $x=e~2.718$ b. $\left(e,\infty \right),\left(0,e\right)$
The formula $I\left(t\right)=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{{e}^{t}}$ is the formula for a decaying alternating current.
$t$ | $\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{{e}^{t}}$ |
---|---|
0 | (i) |
$\frac{\pi}{2}$ | (ii) |
$\pi $ | (iii) |
$\frac{3\pi}{2}$ | (iv) |
$2\pi $ | (v) |
$2\pi $ | (vi) |
$3\pi $ | (vii) |
$\frac{7\pi}{2}$ | (viii) |
$4\pi $ | (ix) |
[T] The population of Toledo, Ohio, in 2000 was approximately 500,000. Assume the population is increasing at a rate of 5% per year.
a. $P=\mathrm{500,000}{\left(1.05\right)}^{t}$ individuals b. ${P}^{\prime}\left(t\right)=24395\xb7{\left(1.05\right)}^{t}$ individuals per year c. $\mathrm{39,737}$ individuals per year
[T] An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present.
[T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1961 is modeled by the function
$N\left(t\right)=5.3{e}^{0.093{t}^{2}-0.87t},(0\le t\le 4),$
where $N\left(t\right)$ gives the number of cases (in thousands) and t is measured in years, with $t=0$ corresponding to the beginning of 1960.
a. At the beginning of 1960 there were 5.3 thousand cases of the disease in New York City. At the beginning of 1963 there were approximately 723 cases of the disease in the United States. b. At the beginning of 1960 the number of cases of the disease was decreasing at rate of $\mathrm{-4.611}$ thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of $\mathrm{-0.2808}$ thousand per year.
[T] The relative rate of change of a differentiable function $y=f\left(x\right)$ is given by $\frac{100\xb7{f}^{\prime}\left(x\right)}{f\left(x\right)}\text{\%}.$ One model for population growth is a Gompertz growth function, given by $P\left(x\right)=a{e}^{\text{\u2212}b\xb7{e}^{\text{\u2212}cx}}$ where $a,b,$ and $c$ are constants.
For the following exercises, use the population of New York City from 1790 to 1860, given in the following table.
Years since 1790 | Population |
---|---|
0 | 33,131 |
10 | 60,515 |
20 | 96,373 |
30 | 123,706 |
40 | 202,300 |
50 | 312,710 |
60 | 515,547 |
70 | 813,669 |
[T] Using a computer program or a calculator, fit a growth curve to the data of the form $p=a{b}^{t}.$
$p=35741{\left(1.045\right)}^{t}$
[T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.
[T] Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year.
Years since 1790 | $P\text{\u2033}$ |
---|---|
0 | 69.25 |
10 | 107.5 |
20 | 167.0 |
30 | 259.4 |
40 | 402.8 |
50 | 625.5 |
60 | 971.4 |
70 | 1508.5 |
[T] Using the tables of first and second derivatives and the best fit, answer the following questions:
True or False ? Justify the answer with a proof or a counterexample.
A continuous function has a continuous derivative.
If a function is differentiable, it is continuous.
Use the limit definition of the derivative to exactly evaluate the derivative.
$f\left(x\right)=\frac{3}{x}$
Find the derivatives of the following functions.
$f\left(x\right)=3{x}^{3}-\frac{4}{{x}^{2}}$
$9{x}^{2}+\frac{8}{{x}^{3}}$
$f\left(x\right)={\left(4-{x}^{2}\right)}^{3}$
$f\left(x\right)={e}^{\text{sin}\phantom{\rule{0.1em}{0ex}}x}$
${e}^{\text{sin}\phantom{\rule{0.1em}{0ex}}x}\text{cos}\phantom{\rule{0.1em}{0ex}}x$
$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(x+2\right)$
$f\left(x\right)={x}^{2}\text{cos}\phantom{\rule{0.1em}{0ex}}x+x\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}\left(x\right)$
$x\phantom{\rule{0.1em}{0ex}}{\text{sec}}^{2}\left(x\right)+2x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(x\right)+\text{tan}\phantom{\rule{0.1em}{0ex}}\left(x\right)-{x}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)$
$f\left(x\right)=\sqrt{3{x}^{2}+2}$
$f\left(x\right)=\frac{x}{4}\phantom{\rule{0.1em}{0ex}}{\text{sin}}^{\mathrm{-1}}\left(x\right)$
$\frac{1}{4}\left(\frac{x}{\sqrt{1-{x}^{2}}}+{\text{sin}}^{\mathrm{-1}}\left(x\right)\right)$
${x}^{2}y=\left(y+2\right)+xy\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)$
Find the following derivatives of various orders.
First derivative of $y=x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x$
$\text{cos}\phantom{\rule{0.1em}{0ex}}x\xb7\left(\text{ln}\phantom{\rule{0.1em}{0ex}}x+1\right)-x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x$
Third derivative of $y={\left(3x+2\right)}^{2}$
Second derivative of $y={4}^{x}+{x}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)$
${4}^{x}{\left(\text{ln}\phantom{\rule{0.1em}{0ex}}4\right)}^{2}+2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+4x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x-{x}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}x$
Find the equation of the tangent line to the following equations at the specified point.
$y={\text{cos}}^{\mathrm{-1}}\left(x\right)+x$ at $x=0$
Draw the derivative for the following graphs.
The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by $w\left(t\right)=1.9+2.9\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(\frac{\pi}{6}t\right),$ where t is measured in hours after midnight, and the height is measured in feet.
Find and graph the derivative. What is the physical meaning?
Find ${w}^{\prime}\left(3\right).$ What is the physical meaning of this value?
${w}^{\prime}\left(3\right)=-\frac{2.9\pi}{6}.$ At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.
The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.
Hours after Midnight, August 26 | Wind Speed (mph) |
---|---|
1 | 45 |
5 | 75 |
11 | 100 |
29 | 115 |
49 | 145 |
58 | 175 |
73 | 155 |
81 | 125 |
85 | 95 |
107 | 35 |
Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?
Estimate the derivative of the wind speed at hour 83. What is the physical meaning?
$\mathrm{-7.5}.$ The wind speed is decreasing at a rate of 7.5 mph/hr
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