By the Mean 4.2 Mean Value Theorem 255 7. f( x ) = 16;[- 2, 2] 23. f (x ) =-2x ifx 0 and for x 0. 11-18. Rolle's Theorem Determine whether Rolle's Theorem applies c. Prove that tan 1 x + tan (1/x) = -w/2, for x 0 so use- guaranteed to exist by Rolle's Theorem. 1- if x 0 36. Without evaluating derivatives, determine which of the functions 19. Lapse rates in the atmosphere Refer to Example 2. Concurrent f(x) = In x, g(x) = In 2x, h(x) = In x2, and p(x) = In 10x2 measurements indicate that at an elevation of 6.1 km, the tempera- have the same derivative. ture is - 10.3 C, and at an elevation of 3.2 km, the temperature is 3.0 C. Based on the Mean Value Theorem, can you conclude that 37. Without evaluating derivatives, determine which of the functions the lapse rate exceeds the threshold value of 7'C/km at some in- s(x) = 2x , h(x) = x10 + 2, and p(x) = x10 - In 2 have the termediate elevation? Explain. same derivative as f(x) = x10. 20. Drag racer acceleration The fastest drag racers can reach a speed 38. Find all functions f whose derivative is f' (x) = x + 1. of 330 mi/ hr over a quarter-mile strip in 4.45 seconds (from a 39. Mean Value Theorem and graphs By visual inspection, locate standing start). Complete the following sentence about such a drag all points on the interval (-4, 4) at which the slope of the tangent racer: At some point during the race, the maximum acceleration of line equals the average rate of change of the function on the inter- the drag racer is at least -mi/hr/s. val [-4, 4]. 21-32. Mean Value Theorem Consider the following functions on the given interval [a, b]. a. Determine whether the Mean Value Theorem applies to the follow- ing functions on the given interval [a, b]. b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 21. f ( x ) = 7 - x2; [ - 1, 2] 22. f (x) = x3 - 2x2; [0, 1] - 2 +