11. Set f(x) = (e^x + e^−x)³. Calculate the rate of change of f with respect to x.

12. Find x as a function of z when z²Â Â· e^x = 8. Then calculate the rate of change of x with respect to z.

13. Find x as a function of z when z³Â Â· e^x = 2. Then calculate the rate of change of x with respect to z.

14. Find x as a function of y when y = y²Â + lnx. Then calculate the rate of change of x with respect to y.

15. Find the slope of the line tangent to y = 5x at x = 1.

16. Find the slope of the line tangent to y = e^x + 3x at x = 0.

17. Find the maximum value of f(x) = ln(9 − x²). Suggestion: first consider the domain of f.

18. Find the maximum value of f(x) = x²e^{x^2Â} .

19. Find the maximum value of g(u) = ueu.

20. Find the maximum value of f(x) = ln(1+x − x²) one the interval [2, 6].

21. The value of a certain machine at time t is V = 50, 000e^−0.08t dollars. (Here t is measured in years and t ≥ 0). At what time will the machine have a value of $10,000?

22. The value of a certain machine at time t is V = 50, 000e^−0.08t dollars. (Here t is measured in years and t ≥ 0). At what rate is the value changing over time?

23. Recall that the probability density function, f(x) of a stochastic variable X whose cumulative distribution function is G(x) = P(X ≤ x) is given by f = dG/dx. A variable is uniformly distributed in [0, 3] when G(x) = 0 for x < 0, G(x) = x/3 for 0 ≤ x < 3, and G(x) = 1 for x ≥ 3. Calculate the probability density function for this variable. Explain why it makes sense to set f(x) = 0 when x < 0 and when x > 3.

24. The population of a certain country, in millions, at time t years is P(t) = 55e^0.03t. At what rate is the population increasing with time? What is the population's relative rate of growth?

25. The capital in a certain country, in billions of dollars, at time t years is K(t) = 800e^0.02t. At what rate is capital increasing with time? What is the relative rate of growth of capital?

26. The total income in a certain country, in billions of dollars, at time t years is I(t) = 20e^0.025t. At what rate is the income increasing with time? In the same country the population, in millions, is P(t) = 55e^0.03t. Calculate the personal income W(t) in this country. Is personal income increasing over time?

27. The population of a certain country, in millions, at time t years is P(t) = 50e^0.03t. The capital, in billions of dollars, is K(t) = 800e^0.02t. Does the relative rate of growth of capital exceed the population's relative rate of growth?

28. The population of a certain country, in millions, at time t years is P(t) = 12e^0.015t. The capital, in billions of dollars, is K(t) = 800e^0.03t. Suppose that personal income is W(t) = 0.01K(t)/P(t). Is personal income increasing over time?

29. Suppose that a painting has the market value of 600(1 + 0.07t²), where time, t, is measured in years since its creation. Assume the discount rate is 0.04 (4 percent). Calculate the present value, v(t), of the painting at time t. At what rate is v(t) changing? Is the present value ever maximized? and if so, at what time?

30. Suppose that a painting has the market value of 300(1 + 0.5t), where time, t, is measured in years since its creation. Assume the discount rate is 0.02 (2 percent). Calculate the present value, v(t), of the painting at time t. At what rate is v(t) changing? Is the present value ever maximized? and if so, at what time?

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Status NEW Posted 13 Aug 2017 09:08 AM My Price 8.00

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