38. Writing g for the acceleration due to gravity, the period, T, of a pendulum of length I is given by T = 271 g a. Show that if the length of the pendulum changes by Al, the change in the period, AT, is given by AT ~ 21 Al. b. If the length of the pendulum increases by 2%, by what percent does the period change?22. a. Find the tangent line approximation near x = 0 to f(x) = 1/(1 - x). b. Use it to approximate 1/0.99.18. a. Show that 1+kX is the local linearization of [1 +X}k nearx = 0. b. Someone claims that the square root of 1.1 is about 1.05. Without using a calculator; do you think that this estimate is about right? C. Is the actual number above or below 1.05? 12. a. Find the best linear approximation, L(x), to f(x) = e* near x = 0. b. What is the sign of the error, E(x) = f(x) - L(x) for x near 0? c. Find the true value of the function at x = 1. What is the error? (Give decimal answers.) Illustrate with a graph. d. Before doing any calculations, explain which you expect to be larger, E(0.1) or E(1), and why. e. Find E(0.1).\f42. For constants a, b, n, R, Van der Waal's equation relates the pressure, P, to the volume, V, of a fixed quantity of a gas at constant temperature T: (P + 1/ 2 ( V - nb ) = nRT. Find the rate of change of volume with pressure, dV/dP.\fI In Exercises 15, decide if the statements are true or false. Give an explanation for your answer. 4. The Racetrack Principle can be used to justify the statement that iftwo horses start a race at the same time, the horse that wins must have been moving faster than the other throughout the race. In Exercises 6-9, does the function graphed appear to satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? Does it satisfy the conclusion? f(x) X 6. a bf (x) X a b 8.16. Applying the Mean Value Theorem with a = 2, b = 7 to the function in Figure 3.46 leads to c = 4. What is the equation of the tangent line at 4? (7,9) (4, 8) / (2, 5) Figure 3.4620. Use the Racetrack Principle and the fact that sin 0 = 0 to Show that sin x S X for all X E 0. 32. Prove that if f (x) = g'(x) for all x in (a, b), then there is a constant ( such that f(x) = g(x) + Con (a, b). [Hint: Apply the Constant Function Theorem to h(x) = f(x) - g(x).]