FINAL REVIEW (2) Compute the area of the region between the vertical lines x = 0 and x = 1, and between the x-axis and the graph of y = x3 . (1) True or false (a) If f and g are differentiable, then d [f (x) + g(x)] = f 0 (x) + g 0 (x). dx (b) If f is differentiable, then f 0 (x) dp f (x) = p . dx 2 f (x) (c) If f and g are differentiable, then (3) Find lim d [f (x)g(x)] = f 0 (x)g 0 (x). dx x→1+ (d) If f and g are continuous on [a, b], then Z b Z b Z b f (x)g(x)dx = f (x)dx g(x)dx . a a x2 −9 x2 +2x−3 (4) Find the absolute maximum and absolute minimum values of f (x) = x4 −2x2 + 3 on [−2, 3]. a (e) If f is continuous on [a, b], then Z b Z b xf (x)dx = x f (x)dx. a a (5) Find the area of the region bounded by the curves y = x2 − x − 6 and y = 0 (f) If f and g are continuous and f (x) ≥ g(x) for a ≤ x ≤ b, then Z b Z b f (x)dx ≥ g(x)dx. a a 1 (6) Find the points on the graph y = x2 at which the tangent line is parallel to the line y = 6x − 1. (8) Find the volume of the solid obtained by rotating the region bounded by x2 −y 2 = a2 and x = a + h about the y-axis (where a and h are positive constants). (7) Find the local extrema of f (x) = 2x6 − 6x4 . Find the intervals on which the graph of f is concave up or concave down, and find the x-coordinates of the points of inflection. Sketch the graph. (9) Find lim v→4+ 4−v |4−v| (10) Find an antiderivative for f (x) = (11) Find the derivative of xecot x (12) Evaluate 2 R sin π+x 5 dx √ 1 1−x2 (16) Find the area of the region bounded by the curves y = ex − 1, y = x2 − x, and x = 1. (13) Find the length of the curve 3 1 y = (x2 + 4) 2 6 for 0 ≤ x ≤ 3. (17) If y = x3 + 2x and x = 2. (14) Find lim x→∞ dx dt = 5, find dy dt when √ ln x x+ln x (18) The altitude of a triangle is increasing at a rate of 1cm/min while the area of the triangle is increasing at a rate of 2cm2 /min. At what rate is the base of the triangle changing when the altitude is 10cm and the area is 100cm2 ? (15) Find the local extrema of f (x) = x ln x. Determine where f is increasing or decreasing. Discuss the concavity, find the points of inflection, and sketch the graph of f . 3 1−2x2 −x4 4 x→−∞ 5+x−3x (b) The y-axis (19) Find lim (20) Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by y = x3 and y = x2 about y = 1. (c) y = 2 (24) Find the absolute maximum and absolute minimum values of f (x) = xe− [−1, 4] (21) Differentiate h(θ) = etan 2θ (22) Evaluate R1 0 (1 − x9 )dx 1 (23) Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x2 about the following lines: (a) The x-axis (25) Find the derivative of f (x) = (ln x) 5 4 x2 8 , (26) Two cars start moving from the same point. One travels south at 60mi/h and the other travels west at 25mi/h. At what rate is the distance between the cars increasing two hours later? (29) Find the equation for the line tangent to 8 the curve y = 4+x 2 at the point (2, 1) (30) Evaluate R1 0 y(y 2 + 1)5 dy (31) What is the 99th derivative of xex ? 1 (27) Differentiate y = xe x (28) Find the volume of the solid obtained by rotating the region bounded by y = 2x and y = x2 about the x-axis. t3 3 t→0 tan 2t (32) Find lim (33) Evaluate 5 R1 0 eπt dt √ R √ 3 (38) Compute ( x− x4 + (34) Find the derivative of f (x) = xln x (35) if A is a positive constant, then find e2x +ex lim Ae 2x −ex 7 √ 3 2 x −6ex +1)dx (39) Find the volume of the solid obtained by rotating the region bounded by x = 0 and x = 9 − y 2 about x = −1. x→−∞ (36) √ Determine the area under the curve y = a2 − x2 and between the lines x = 0 and x = a. 2x x −2x 3 x→∞ (40) Find lim (41) Evaluate (37) Each side of a square is increasing at a rate of 8cm/s. At what rate is the area of the square increasing when that area of the square is 16cm2 ? 6 R √ x+2 dx x2 +4x (42) Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Find all numbers c that satisfy the conclusion of the Mean Value Theorem. (44) RUse a sin 2x √ substitution dx 1+cos 2x to evaluate f (x) = e−2x , [0, 3] ex −x2 2x +e−x e x→∞ (45) Find lim (43) Find the local extrema of f (x) = 5 − 7x − 4x2 and the intervals on which f is increasing or is decreasing, and sketch a graph of f . (46) Find the extrema and sketch the graph 2 of f (x) = x +2x−8 x+3 7 √ (47) Evaluate R e√ x dx x (49) Differentiate H(v) = v arctan v (50) The interior of a half-mile track consists of a rectangle with two semicircles at two opposite ends. Find the dimensions that will maximize the area of the rectangle. (48) A man wishes to put a fence around a rectangular field and then subdivide this field into three smaller rectangular plots by placing two fences parallel to one of the sides. If he can afford only 1000 yards of fencing, what dimensions will give him the maximum area? √ x √e x x→∞ e +1 (51) Find lim (52) Evaluate R x3 dx 1+x4 (53) Find the derivative of f (x) = x2 ex 8 Answers 9 (22) 10 (23) (a) 2π 15 (b) π6 (c) 8π 15 (24) f (2) = (1) True or false (a) True (b) True (c) False (d) False (e) False. (f) True. (2) 0.25 (3) −∞ (4) f (3) = 66, f (±1) = 2 (5) 125 6 (6) Since the slope of y = 6x − 1 is 6, the slope of the tangent line must be 6. Thus the derivate 2x = 6, x = 3. The desired point is (3, 9). √ (7) f (0) = 0 is a maximum, f (± 2) = −8) are q minima, qconcave up on (−∞, − 6 5) and ( q q down on (− 65 , 65 ). (8) (9) (10) (11) (12) (13) (14) (15) 4 3 π(2ah 6 5 , ∞), (25) f 0 (x) √2 , e f (−1) = − 1 = 1 1 e8 4 5x(ln x) 5 (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) concave 3 + h2 ) 2 −1 arcsin x ecot x (1 − x csc2 x) −5 cos π+x 5 +C 15 2 0 Min: f (e−1 ) = −e−1 , increasing on [e−1 , ∞), decreasing on (0, e−1 ), concave up on (0, ∞), no points of inflection. (16) e − 11 6 (17) 70 (18) −1.6cm/min (19) 31 R1 (20) 0 π[(1 − x3 )2 − (1 − x2 )2 ]dx (21) h0 (θ) = 2 sec2 (2θ)etan 2θ (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) 9 65mi/h. 1 y 0 = e− x (1 + x1 ) 64π 15 y = − 12 x + 2 21 4 xex + 99ex 1 8 1 π π (e − 1) 2xln x−1 ln x −1 πa2 4 2 48cm √ /s 3 √ √ 3 2 x 3 3 7 3 x − 7 x + 21 x − 6e + x + C 1656π 5 0 √ x2 + 4x + C c = − 12 ln( 61 (1 − e−6 )) Max: f (− 78 ) = 129 16 , increasing on 7 (−∞, − 8 ), decreasing on [− 87 , ∞) √ − 1 + cos 2x + C 0 no√extrema 2e x + C 125 yards by 250 yards v H 0 (v) = 1+v 2 + arctan v 1 radius is 8π mi and length is 18 mi. 1 1 4 4 ln |1 + x | + C f 0 (x) = x(x + 2)ex