미분법 전체 문제

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)=\log (\sec x)$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)={\log }_{7}{(6{x}^{4}+3)}^{5}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)={2}^{x}\cdot {\log }_{3}{7}^{{x}^{2}-4}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={x}^{\sqrt{x}}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={(\ln x)}^{\ln x}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={x}^{{\log }_{2}x}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={({x}^{2}-1)}^{\ln x}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={x}^{\cot x}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y=\frac{x+11}{\sqrt[3]{{x}^{2}-4}}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={x}^{-1/2}{({x}^{2}+3)}^{2/3}{(3x-4)}^{4}$

[T] Find an equation of the tangent line to the graph of $f(x)=4x{e}^{({x}^{2}-1)}$ at the point where

$x=-1.$ Graph both the function and the tangent line.

[T] Find an equation of the tangent line to the graph of ${x}^{3}-x\ln y+{y}^{3}=2x+5$ at the point (2, 1). (Hint: Use implicit differentiation to find $\frac{dy}{dx}.)$ Graph both the curve and the tangent line.

Consider the function $y={x}^{1/x}$ for $x>0.$

a. Determine the points on the graph where the tangent line is horizontal. b. Determine the points on the graph where ${y}^{'}>0$ and those where ${y}^{'}<0.$

The formula $I(t)=\frac{\sin t}{{e}^{t}}$ is the formula for a decaying alternating current.

a. Complete the following table with the appropriate values.

[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. Write the exponential function that relates the total population as a function of $t.$ b. Use a. to determine the rate at which the population is increasing in $t$ years. c. Use b. to determine the rate at which the population is increasing in 10 years.

[T] An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present.

a. Write the exponential function that relates the amount of substance remaining as a function of $t,$ measured in hours. b. Use a. to determine the rate at which the substance is decaying in $t$ hours. c. Use b. to determine the rate of decay at $t=4$ hours.

[T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1964 is modeled by the function

$N(t)=5.3{e}^{0.093{t}^{2}-0.87t},(0\le t\le 4),$ where $N(t)$ gives the number of cases (in thousands) and t is measured in years, with $t=0$ corresponding to the beginning of 1960.

a. Show work that evaluates $N(0)$ and $N(4).$ Briefly describe what these values indicate about the disease in New York City. b. Show work that evaluates ${N}^{'}(0)$ and ${N}^{'}(3).$ Briefly describe what these values indicate about the disease in New York City.

[T] The relative rate of change of a differentiable function $y=f(x)$ is given by $\frac{100\cdot {f}^{'}(x)}{f(x)}\%.$ One model for population growth is a Gompertz growth function, given by $P(x)=a{e}^{-b\cdot {e}^{-cx}}$ where $a,b,$ and $c$ are constants.

a. Find the relative rate of change formula for the generic Gompertz function. b. Use a. to find the relative rate of change of a population in $x=20$ months when $a=204,b=0.0198,$ and $c=0.15.$ c. Briefly interpret what the result of b. means.

[T] Using a computer program or a calculator, fit a growth curve to the data of the form $p=a{b}^{t}.$

[T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.