For the following exercises (1-15), find [latex]f^{\prime}(x)[/latex] for each function.
1. [latex]f(x)=x^2 e^x[/latex]
2. [latex]f(x)=\dfrac{e^{−x}}{x}[/latex]
3. [latex]f(x)=e^{x^3 \ln x}[/latex]
4. [latex]f(x)=\sqrt{e^{2x}+2x}[/latex]
5. [latex]f(x)=\dfrac{e^x-e^{−x}}{e^x+e^{−x}}[/latex]
6. [latex]f(x)=\dfrac{10^x}{\ln 10}[/latex]
7. [latex]f(x)=2^{4x}+4x^2[/latex]
8. [latex]f(x)=3^{\sin 3x}[/latex]
9. [latex]f(x)=x^{\pi} \cdot \pi^x[/latex]
10. [latex]f(x)=\ln(4x^3+x)[/latex]
11. [latex]f(x)=\ln \sqrt{5x-7}[/latex]
12. [latex]f(x)=x^2 \ln 9x[/latex]
13. [latex]f(x)=\log(\sec x)[/latex]
14. [latex]f(x)=\log_7 (6x^4+3)^5[/latex]
15. [latex]f(x)=2^x \cdot \log_3 7^{x^2-4}[/latex]
For the following exercises (16-23), use logarithmic differentiation to find [latex]\frac{dy}{dx}[/latex].
16. [latex]y=x^{\sqrt{x}}[/latex]
17. [latex]y=(\sin 2x)^{4x}[/latex]
18. [latex]y=(\ln x)^{\ln x}[/latex]
19. [latex]y=x^{\log_2 x}[/latex]
20. [latex]y=(x^2-1)^{\ln x}[/latex]
21. [latex]y=x^{\cot x}[/latex]
22. [latex]y= \dfrac{x+11}{\sqrt[3]{x^2-4}}[/latex]
23. [latex]y=x^{-\frac{1}{2}}(x^2+3)^{\frac{2}{3}}(3x-4)^4[/latex]
24. [T] Find an equation of the tangent line to the graph of [latex]f(x)=4xe^{x^2-1}[/latex] at the point where
[latex]x=-1[/latex]. Graph both the function and the tangent line.
25. [T] Find the equation of the line that is normal to the graph of [latex]f(x)=x \cdot 5^x[/latex] at the point where [latex]x=1[/latex]. Graph both the function and the normal line.
26. [T] Find the equation of the tangent line to the graph of [latex]x^3-x \ln y+y^3=2x+5[/latex] at the point where [latex]x=2[/latex]. Graph both the curve and the tangent line.
27. Consider the function [latex]y=x^{\frac{1}{x}}[/latex] for [latex]x>0[/latex].
- Determine the points on the graph where the tangent line is horizontal.
- Determine the intervals where [latex]y^{\prime}>0[/latex] and those where [latex]y^{\prime}<0[/latex].
28. The formula [latex]I(t)=\dfrac{\sin t}{e^t}[/latex] is the formula for a decaying alternating current.
- Complete the following table with the appropriate values.
[latex]t[/latex] [latex]\frac{\sin t}{e^t}[/latex] 0 (i) [latex]\frac{\pi}{2}[/latex] (ii) [latex]\pi[/latex] (iii) [latex]\frac{3\pi}{2}[/latex] (iv) [latex]2\pi[/latex] (v) [latex]2\pi[/latex] (vi) [latex]3\pi[/latex] (vii) [latex]\frac{7\pi}{2}[/latex] (viii) [latex]4\pi[/latex] (ix) - Using only the values in the table, determine where the tangent line to the graph of [latex]I(t)[/latex] 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.
- Write the exponential function that relates the total population as a function of [latex]t[/latex].
- Use a. to determine the rate at which the population is increasing in [latex]t[/latex] years.
- 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.
- Write the exponential function that relates the amount of substance remaining as a function of [latex]t[/latex], measured in hours.
- Use a. to determine the rate at which the substance is decaying in [latex]t[/latex] hours.
- Use b. to determine the rate of decay at [latex]t=4[/latex] 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
[latex]N(t)=5.3e^{0.093t^2-0.87t}, \, (0\le t\le 4)[/latex],
where [latex]N(t)[/latex] gives the number of cases (in thousands) and [latex]t[/latex] is measured in years, with [latex]t=0[/latex] corresponding to the beginning of 1960.
- Show work that evaluates [latex]N(0)[/latex] and [latex]N(4)[/latex]. Briefly describe what these values indicate about the disease in New York City.
- Show work that evaluates [latex]N^{\prime}(0)[/latex] and [latex]N^{\prime}(3)[/latex]. Briefly describe what these values indicate about the disease in New York City.
32. [T] The relative rate of change of a differentiable function [latex]y=f(x)[/latex] is given by [latex]\frac{100 \cdot f^{\prime}(x)}{f(x)}\%[/latex]. One model for population growth is a Gompertz growth function, given by [latex]P(x)=ae^{−b \cdot e^{−cx}}[/latex] where [latex]a, \, b[/latex], and [latex]c[/latex] are constants.
- Find the relative rate of change formula for the generic Gompertz function.
- Use a. to find the relative rate of change of a population in [latex]x=20[/latex] months when [latex]a=204,b=0.0198,[/latex] and [latex]c=0.15.[/latex]
- Briefly interpret what the result of b. means.
For the following exercises (33-36), 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 |
33. [T] Using a computer program or a calculator, fit a growth curve to the data of the form [latex]p=ab^t[/latex].
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:
- Will the model be accurate in predicting the future population of New York City? Why or why not?
- Estimate the population in 2010. Was the prediction correct from a.?