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

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

25. [T] Find the equation of the line that is normal to the graph of $f(x)=x \cdot 5^x$ at the point where $x=1$. Graph both the function and the normal line.

Show Solution

$y=\frac{-1}{5+5 \ln 5}x+(5+\frac{1}{5+5 \ln 5})$

26. [T] Find the equation of the tangent line to the graph of $x^3-x \ln y+y^3=2x+5$ at the point where $x=2$. Graph both the curve and the tangent line.

27. Consider the function $y=x^{\frac{1}{x}}$ for $x>0$.

1. Determine the points on the graph where the tangent line is horizontal.
2. Determine the intervals where $y^{\prime}>0$ and those where $y^{\prime}<0$.

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

1. Complete the following table with the appropriate values.
   

   $t$	$\frac{\sin t}{e^t}$
   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)

2. Using only the values in the table, determine where the tangent line to the graph of $I(t)$ is horizontal.

29. [T] The population of Toledo, Ohio, in 2000 was approximately 500,000. Assume the population is increasing at a rate of 5% per year.

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

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

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

31. [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

$N(t)=5.3e^{0.093t^2-0.87t}, \, (0\le t\le 4)$,

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.

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

32. [T] The relative rate of change of a differentiable function $y=f(x)$ is given by $\frac{100 \cdot f^{\prime}(x)}{f(x)}\%$. One model for population growth is a Gompertz growth function, given by $P(x)=ae^{−b \cdot e^{−cx}}$ where $a, \, b$, and $c$ are constants.

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

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

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

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

36. [T] Using the tables of first and second derivatives and the best fit, answer the following questions:

1. Will the model be accurate in predicting the future population of New York City? Why or why not?
2. Estimate the population in 2010. Was the prediction correct from a.?