student: Practice Exam I Problem 1: Find the derivative of the functions T 1 (x), T 2 (x), T 3 (x). State the reason of your answers. a) T 1 (x) = x 2t dt 2 b) T 2 (x) = e x ln(t2 )dt c) T 3 (x) = x 2 sin(x)ex dx x Problem 2: shown. Let g(x) = x f(t)dt, where f is the function whose graph is Evaluate the function values f() = g (2) = g(1) = g(4) = Where g(x) has a maximal value in the interval [,9]? (Circle the right answer) x = x = 2 x = 3 x = 4 x = 5 x = 6 x = 8 x = 9 On what interval g decreases?
Problem 3: Evaluate the indefinite integrals 3e x + 6x 2 dx sin(x) + 3 x dx Problem 4: Evaluate the definite integral 4 1 x + 1 x dx π/2 π/3 cos(x) dx
Problem 5: Evaluate the definite integrals π/2 sin(x)e cos(x) dx 1 xe 3x2 dx Problem 6: x 2 e 2x dx Evaluate the indefinite integral ln( x) dx
Problem 7: Evaluate the definite integrals 1 e x cos(x) dx 1 e x2 sin(x 2 )x dx Problem 8: Evaluate the indefinite integral x 5 cos(x 3 ) dx xe 2x+1 dx
Problem 9: Use Simpson s Rule to approximate the area under the graph of f(x) in problem 2, bounded by the vertical lines x = 5 and x = 9. Problem 1: Apply the definition of improper integral to decide whether the following integrals are convergent or divergent: 1 x dx 4 2 1 (x 3) 2 dx
Problem 11: Evaluate the indefinite integral x + 1 3x 2 1x + 3 dx Problem 12: u 2 + 3 u 2 u 2 + 1 du = Evaluate the indefinite integral:
Problem 13: Sketch the region bounded by the curves: y = x 2 and x = y 2 Label the curves and determine any points of intersection. Then find the area of this region. Problem 14: Sketch the region bounded by the curves y = e x, y =,, x =, x = 1. Label the curves and determine any points of intersection. Find the volume of the solid obtained by rotating this region about the x-axis. Write down an integral expression for the volume of the solid obtained by rotating the above region about the y-axis. (Do NOT try to evaluate the last integral.)
Problem 15: Graph the arc of the curve C and find its exact length, where C is given by the parametric equations: x = t t 2, y = t 2 + 2 t + 2, 1 t 4 Problem 16: Let f(x) = 4 x 2 for in the interval [,2]. Find the average value of f on the given interval. Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f. Find a number c, such that the average value of f is f(c).
Problem 17: Probably something from Applications of Integration I. Problem 18: Probably something from Applications of Integration I.
Problem 19: Might be chosen from Concept Check questions on page 438. Problem 2: Might be chosen from Concept Check questions on page 499.
1: 2: 3: 4: 5: 6: 7: 8: 9: 1: 11: 12: 13: TABLE OF INTEGRALS a2 + u 2 du = u a2 + u 2 2 + a2 ln(u + a 2 2 + u 2 ) + C du u = ln(u + a 2 + u 2 ) + C 2 +a 2 du u 2 = a2 +u 2 + C u 2 +a 2 a 2 u a 2 +u 2 du = a 2 +u 2 u 2 u du a 2 +u 2 u 2 du u 2 a 2 = u 2 = ln(u + a 2 + u 2 ) + C + ln(u + a 2 + u 2 ) + C u2 a 2 + a2 2 ln u + u 2 a 2 + C u 2 du u a 2 = 2 15 (8a4 + 3u 2 + 4a 2 u) u a 2 + C sin(au) cos(bu) du = cos((a b)u) 2(a b) cos((a+b)u) 2(a+b) + C u 2 du a = u a2 + u 2 +u 2 2 2 a2 ln(u + a 2 2 + u 2 ) + C u cos(u) du = cos(u) + u sin(u) + C e au cos bu du = eau (a cos(bu) + b sin(bu)) + C a 2 +b 2 ue au du = 1 (au 1)e au + C a 2 du u = 1 a+ ln( a 2 +u 2 ) + C a 2 +u 2 a u Values of trigonometric functions, you may find useful: sin() = sin π 2 = 1 sin π = sin π 3 = 3 2 cos π 6 = 3 2 cos π 3 = 1 2