8. Consider the basic functions f(x) = a and g(x) = sin(x). a. Let h(x) = f(g(x)). Find the exact instantaneous rate of change of h at the point where ~ = b. Which function is changing most rapidly at x - 0.25: h(x) = f(g(x)) or r(x) = g(f(x))? Why? c. Let h(x) = f(g(x)) and r(x) = g(f(x)). Which of these functions has a derivative that is periodic? Why?9. Let u(x) be a differentiable function. For each of the following functions, determine the derivative. Each response will involve u and/or u'. a. p(a) = el(z) b. q(z) = u(er) C. r(x) = cot(u(x)) d. s(x) = u(cot(x)) e. a(x) = u(x4) f. b(ac) = u (a)10. Let functions p and q be the piecewise linear functions given by their respective graphs in Figure 2.5.9. Use the graphs to answer the following questions. P - 3 -2 2 3 -2+ 3+ Figure 2.5.9. The graphs of p (in blue) and q (in green). a. Let C(x) = p(q(x)). Determine C'(0) and C'(3). b. Find a value of a for which C'(x) does not exist. Explain your thinking. c. Let Y(x) = q(q(x)) and Z(x) = q(p(x)). Determine Y'(-2) and Z'(0).11. If a spherical tank of radius 4 feet has h feet of water present in the tank, then the volume of water in the tank is given by the formula V = -h-(12 - h). a. At what instantaneous rate is the volume of water in the tank changing with respect to the height of the water at the instant h = 1? What are the units on this quantity? b. Now suppose that the height of water in the tank is being regulated by an inflow and outflow (e.g., a faucet and a drain) so that the height of the water at time t is given by the rule h(t) - sin(nt), + 1, where t is measured in hours (and h is still measured in feet). At what rate is the height of the water changing with respect to time at the instant t = 2? c. Continuing under the assumptions in (b), at what instantaneous rate is the volume of water in the tank changing with respect to time at the instant t = 2? d. What are the main differences between the rates found in (a) and (c)? Include a discussion of the relevant units