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5. Answer each of the following either TRUE or FALSE. (a) Let f and g be any two functions which are continuous on [0, 1), with f(0) = g(0) = 0 and /(1) = g(1) = 10. Then there must exist c, de [0, 1] such that f'(c) = g'(d). (b) Let f and g be any two functions which are continuous on [0, 1] and differentiable on (0, 1), with f(0) = g(0) = 0 and f(1) = 9(1) = 10. Then there must exist ce [0, 1] such that /(c) = g'(c). (c) For all r in the domain of sec-1 r, sec( sec -' (r)) = r. 6. Answer each of the following either TRUE or FALSE. (a) The slope of the tangent line of f(r) at the point (a, f(a)) is given by f(a th) - f(a) h (b) Using the Intermediate Value Theorem it can be shown that lim r sin - = 0. (c) The graph below exhibits three types of discontinuities. (d) If w = f(x), x = g(y), y = h(=), then du _ dw dr dy de dx dy de (e) Suppose that on the open interval I, f is a differentiable function that has an inverse function f- and /'(x) #0. Then f-1 is differentiable and 1 (5-( ) ) = F(F-(I) for all r in the domain of f-1. (f) Given the graph of f below to the left, the graph to the right must be that of f. (g) The conclusion of the Mean Value Theorem says that the graph of f has at least one tangent line in (a, b), whose slope is equal to the average slope on [a, b].(e) An equivalent precise definition of lim f(r) = [ is: For any 0 0 such that if |x - al 2 (f) f(3) = -1 (@) f(2) = 11 (h) f is continuous at r = 3. (i) f is continuous at r = 2. 8. Answer TRUE or FALSE to the following questions. (a) If a particle has a constant acceleration, then its position function is a cubic polynomial. (b) If f(x) is differentiable on the open interval (a, b) then by the Mean Value Theorem there is a number cin (a, b) such that (b-a) f'(c) = f(b) - f(a). (c) If lim. = 0 for every number k, then lim f(x) = 00. r-+00 (d) If /(x) has an absolute minimum at r = c, then /'(c) = 0. 9. True or False. Give a brief justification for each answer. 87 (a) There is a differentiable function f(r) with the property that /(1) = -2 and f(5) = 14 and f'(r) + 2, C. y =x - 2, D. y= > -4, E. none of the above 15. This is a multiple choice question. No explanation is required (a) The derivative of g(r) = eve is A. Vxevi-1, B. 2eViz-0.5 0.5evi Vi : D. eva, E. None of these (b) If coshy = r + ray, then at the point (1,0) wey = A. 0, B. - 1, C. 1, D. 3, E. Does not exist (c) An antiderivative of f(x) = r - sinc ter is A. 1 - cosa tex, B. x' + Inx - cost, C. 0.5x- + e - cost, D. cost + er + 0.5r', E. None of these 91 (d) If h(x) = In(1 - x?) where - 1