31. PRACTICE PROBLEMS Below is a collection of problems from all chapters in no particular order. (1) Evaluate the following limits: JE+1, 1# 2 (a) lim f(x), where f(I) = VT, I = 2, (b) lim t-+0 1 2 + t 2x-6 + 45x4 + 5 lim x-+00 316 + 50012 + 100x + 2000 (2) For what values of c is the given function continuous at r = 3? C2 - 2 2 f(I) =12 - 5x +6 x > 3 x - 3 (3) Evaluate the integral: e sin(eF) dx. (4) (a) Using the definition of the derivative, find the derivative of f(I) = > +, 1 (b) State the domain of f and f'. (5) Sketch the graph of a function that satisfies all of the given conditions. . f'(x) > 0, for all x # 1 . x = 1 is a vertical asymptote . f" > 0, for I E (-0o, 1) U(3, co) . x = 3 is an inflection point. (6) Prove that the equation x' + r - 1 = 0 has exactly one real root. (Hint: Rolle's Theorem). (7 Find the derivative of the function: y = / V1 + +2dt. (8) A particle moves with velocity function v(t) = sint - cost and its initial displacement is s(0) = 0 m. Find its position function after t seconds. (9) Let f(x) = 1_ 2 with derivatives f'(x) = 2x (1-22)2, and f"(x) = _ 6x2 + 2 (1- 22)3 For the function f, determine: (a) The vertical and horizontal asymptotes. (b) The intervals of increase and decrease. (c) The local maximum and minimum values. (d) The intervals of concavity and inflection points. (e) Use the information from above to sketch f. (10) Evaluate the integral