2x (b) f" (2) = - (1 + x2)2 + sin(2), f'(0) = 2 and f(0) = -1, find f (x) . (4) (4 marks) In questions 4a-4b, find the value or a formula for the given sum. (a) (i - 1) (2+ 2). i= 1 199 (b) sin(,)(3) (4 marks) In questions 3a-3b, use the given information to find the function or its values at the given point (i.e., solve the Initial Value Problem). (a) f'(a) = 203 - 2 2-5 2 = 0, find f (x) .3 (5) (3 marks) In this question, first write the sum in Sigma notation as the Riemann sum for a function and express the limit as a definite integral / f(x)dx. Then evaluate the integral. (n +2 ) 12 12 lim BIN + (n + 4 ) + 2 +6) 12 + ...+ (n+ 2n ) 12 n-con n n n n (6) (4 marks) In questions 6a-6b, evaluate the given integral using properties of definite integrals and substitution. (a) Let f(x)dx = 3 and 2f(x)dx = 4 and 3f(x)dx = -3. Find fo f(x)dx. (b) Let / xch(x2) dx = 3. Find ffh(x) dx.(7) (4 marks) In questions 7a-7b, nd the derivative. d :c sin(t) dt = (a) a; [0 e (8) (6 marks) An object moves along a line with velocity function given by v(t) = t2 4t + 3. (a) Find the displacement of the particle during 0 S t S 6. (b) Find the distance traveled by particle during 0 g t S 6. 5 (9) (2 marks) Find the interval where the following function g(x) = / et dt is concave up. (10) (3 marks) Find the limit lim Ji t2 In (t) dt 2- 1 fx sin(t - 1) dt(11) (9 marks) In questions 1la-11c, first find the appropriate substitution, change the integral, and evaluate it. 62 + 4x - 1 (a) 2x3 + 2.2 - x + 3 . Appropriate substitution u = . Integral after the substitution is performed= . Evaluate the integral. re2 (b) x (In(x)) 2 dac . Appropriate substitution u = . Integral after the substitution is performed= . Evaluate the integral. (c) 2 5x2 + 4 da . Appropriate substitution u = . Integral after the substitution is performed= . Evaluate the integral