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13.

Find x as a function of z when z3 ? ex = 2. Then calculate the rate of change of x with respect to z.

14. Find x as a function of y when y = y2 + ln x. 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

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

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?

24. The population of a certain country, in millions, at time t years is P(t) = 55e0.03t . 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) = 800e0.02t . At what rate is capital increasing with time? What is the relative rate of growth of capital?

28. The population of a certain country, in millions, at time t years is P(t) = 12e0.015t . The capital, in billions of dollars, is K(t) = 800e0.03t . Suppose that personal income is W(t) = 0.01K(t)/P(t). Is personal income increasing over time?

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