Mathematics Calculus II Summer Session II, 15 Test #1

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1 Mathematics Calculus II Summer Session II, 15 Test #1 Instructor: Dr. Alexandra Shlapentokh (1) Compute the area ounded y the curves y = x and y = x (a) 5 6 () 7 6 (c) 11 6 (d) 1 6 (2) Suppose you have to compute the area ounded y two everywhere continuous graphs y = f(x), y = g(x), and lines x =.1, x =.2. Assume further that for x [0, 1] we have that f(x) > g(x) and the area in question is denoted y A. Which of the statements elow is true? (a) A = () A = (c) A = (g(x) f(x))dx (f(x) + g(x))dx (f(x) g(x))dx (d) There is not enough information to answer this question. () Suppose you have to compute the area ounded y two everywhere continuous graphs y = f(x), y = g(x), and lines x = 8, x = 8. Assume further that for some real numer > 9 it is the case that for x (, 0) we have that f(x) > g(x), and for x (0, ) we have that g(x) > f(x). Then the area in question is equal to (a) () (c) (d) (f(x) g(x))dx (g(x) f(x))dx (f(x) + g(x))dx (f(x) g(x))dx (g(x) f(x))dx () Find the area ounded y y = x + and y = x +. See Figure 1 (a) 1 () 1 (c) 1 1

2 FIGURE FIGURE 2. π/ π/2 (d) 1 2 (5) Find the are etween y = sin x + and y = cos x + for x [0, π/2]. (See Figure 2.) (a) () (c) (d) (6) Find the volume of the figure produced y rotating the region ounded y lines y = x, y = x +, x = 0, x = 5 around x-axis. (See Figure.) 2

3 FIGURE. 5 (a) 200π () 180π (c) 100 (d) 100π (7) Which of the formulas elow will compute the volume of the figure produced y rotating the region ounded y y = 2x and y = (x 2) 2 + around the x-axis. (See Figure.) (a) π 2 0 (x2 2x)dx () π 2 0 [( (x 2)2 ) x ]dx (c) π 2 0 [( (x 2)2 ) 2 + x 2 ]dx (d) π 2 0 [( (x 2)2 ) 2 x 2 ]dx (8) Which of the formulas elow will compute the volume of the solid otained y rotation of the figure ounded y y = 2x and y = (x 2) 2 + around y-axis. (See Figure.) (a) 2 0 2π( (x 2)2 + 2x)dx () 2π 2 0 ( (x 2)2 2x)xdx (c) 2π 2 0 ( (x 2)2 + 2x)xdx (d) π 2 0 ( (x 2)2 2x)xdx (9) Which of the formulas elow will compute the volume of the solid otained y rotation of the figure ounded y y = 2x and y = (x 2) 2 + around the line y = 6. (See Figure.) (a) π 2 0 ((2 + (x 2)2 ) 2 + (6 2x) 2 )dx () π 2 0 ((2 + (x 2)2 ) 2 (6 + 2x) 2 )dx (c) π 2 0 ((2 (x 2)2 ) 2 (6 2x) 2 )dx (d) π 2 0 ((6 2x)2 (2 + (x 2) 2 ) 2 )dx

4 FIGURE. 6 2 FIGURE 5. 2 (10) Find the volume of the figure produced y rotating the region ounded y lines y = x + x and y = 0 for x [0, 2], around y-axis. (See Figure 5.) (a) 128π 5 () 128π 15 (c) 128π (d) 6π 15 (11) Which of the formulas elow computes the volume of the solid resulting from rotating the region ounded y y = x + x and y = x 2 2x around y-axis? (See Figure 6.) (a) 2π () ( x 2x 2 + 6x)xdx ( x x 2 + x)dx

5 FIGURE 6. 2 FIGURE 7. R B r A a h (c) 2π (d) ( x x 2 + 6x)xdx ( x x 2 + 6x)xdx (12) Suppose points A, B lie on a line passing through the origin, the x-coordinate of A is a, and the x-coordinate of B is with a = h. Assume further the y-coordinate of B is R and the y-coordinate of A is r and R > r. See Figure 7. Determine a in terms of h, r, R. (Hint: use similarity of traingles.) (a) a = rh R + r () a = rh (c) a = R 2rh r R r (d) a = 2Rh R r 5

6 (1) Let A, B, a,, R, r, h e as in Prolem 12 and Figure 7. Find the slope of AB. h (a) R r () R + r h (c) R r h (d) r R h (1) Let A, B, a,, R, r, h e as in Prolem 12 and Figure 7. Suppose the slope of AB is m. In this case which of the following formulas computes the the volume of the figure otained y rotating the region ounded y x = a, x =, AB and y = 0 around the x-axis. (a) π () π (c) π (d) π a (m 2 + x 2 )dx m 2 xdx mx 2 dx m 2 x 2 dx (15) Let A, B, a,, R, r, h e as in Prolem 12 and Figure 7. Suppose R = 5, r =, h = 1. In this case what is the the volume of the figure otained y rotating the region ounded y x = a, x =, AB and y = 0 around the x-axis. (a) π () π (c) 7π (d) 9π (16) Find the area ounded y y = x 2 and y = x 2 +. See Region C in Figure 8. (a) 121 () 125 (c) 128 (d) 129 6

7 (17) Find the area ounded y y = x 2, y = x 2 + etween x = and x = 2. See Region A in Figure 8. (a) 61 () 62 (c) 6 (d) 65 (18) The total area ounded y y = x 2, y = x 2 + etween x = and x =, i.e. the total area of Regions A, B, and C in Figure 8, can e computed y the following formula: (a) () (c) (d) 2x dx 2x 2 8 dx 2x 2 16 dx x 2 8 dx (19) The total area ounded y y = x 2, y = x 2 + etween x = and x =, i.e. the total area of Regions A, B, and C in Figure 8, is equal to (a) 256 () 25 (c) 252 (d) 258 (20) What is the formula for computing volume of the solid otained y rotation of a region ounded y graphs y = f(x), y = g(x), x = a, x = around x-axis, if a < are real numers and for all x [a, ] we have that f(x), g(x) are continuous and f(x) > g(x) > 0. (a) π () π (c) π a (f(x) 2 g(x) 2 )dx (f(x) 2 + g(x) 2 )dx (f(x) g(x)) 2 dx 7

8 FIGURE A B C (d) π a (f(x) g(x) )dx (21) What is the volume of the solid otained y rotation of a region ounded y graphs y = f(x), y = g(x), x = 0, x = 1 around x-axis, if f(x), g(x) are continuous, f(x) > g(x) > 0 and f 2 (x) g 2 (x) = 1 for all x [0, 1]. (a) Impossile to determine without more information () π (c) π 2 (d) 2π (22) What is the volume of the solid otained y rotation of a region ounded y graphs y = f(x), y = g(x), x = a, x = around the y-axis, if a < are real numers and for all x [a, ] we have that f(x), g(x) are continuous and f(x) > g(x) > 0. (a) π () 2π (c) 2π (d) 2π a (f(x) 2 g(x) 2 )dx x(f(x) g(x))dx x(f(x) g(x)) 2 dx (f(x) g(x))dx (2) What is the volume of the solid with a circular ase of radius 1 and perpendicular cross sections that are triangles with the height corresponding to the side of the 8

9 FIGURE 9. 1 A 1 x 1 B 1 triangle intersecting the ase of the solid of the same length as the side? (In Figure 9 the unit circle represents the ase and AB is the side of the cross section triangle intersecting the ase at x. The height h of this triangle corresponding to the side AB has the same length as AB.) (a) 8 () 5 (c) 7 (d) 2 (2) What is the work done y the force moving an oject from x = 0 to x = if F (x) = x 2 Newtons and the distance is measured in meters? (a) 7 Newton-meters () 8 Newton-meters (c) 9 Newton-meters (d) 10 Newton-meters (25) What is the average value of the function f(x) = x 2 etween x = 0 and x = if F (x) = x 2? (a) 7/2 () 8 (c) (d) 10 9

10 (26) Suppose the average value of a continuous function f(x) on an interval [a, ] times a is equal to the area ounded y the graph of the function, x-axis, line x = a and line x =. In this case which of the following statements must always e true. (Hint: compare the formula for the average with the formula for the area under the curve.) (a) f(x) 0 for all x [a, ]. () f(x) 0 for all x [a, ]. (c) f(x) 0 for some x [a, ] ut not for all. (d) f(x) = 0 for all x [a, ]. (27) If f(x) is a one-to-one function, then which of the following statements must e always true? (a) f(x) 0 () The range of f consists of all real numers. (c) It is possile to have x 1 x 2 and f(x 1 ) = f(x 2 ). (d) f(1) = 1 (28) Let f(x) = x + 0x x Find (f 1 ) (12). (a) 1 75 () 1 7 (c) 1 72 (d) Cannot e determined (29) Find all the critical points of the function y = xe 2ax 5 and determine their nature. Here a > 0 is a real numer (a) x = 1 : local minimum 2a () x = 1: local maximum (c) x = 1 : local maximum 2a (d) No critical points (0) Find all the critical points of the function y = x + e mx + 0 and determine their nature. Here m > 0 is a real numer. (a) x = 1: local maximum () x = 1 : local maximum m (c) x = 1 : local minimum m (d) No critical points (1) Determine for what values of x the function y = e mx2 + 6x + concaves up, if m is a positive real numer. (a) The graph never concaves up. () The graph concaves up for x 0. (c) The graph concaves up for x 0. 10

11 (d) The graph always concaves up. (2) Determine for what values of x the function y = e mx2 6 is increasing, if m is a positive real numer. (a) The graph is decreasing everywhere. () The graph is increasing everywhere. (c) The graph is increasing for x > 0. (d) The graph is increasing for x < 0. () Find the f 1 (x) if f(x) = x 1 x+1. (a) f 1 (x) = x2 + 1 x 1 () f 1 (x) = x + 1 x 2 1 (c) f 1 (x) = x x (d) f 1 (x) = 2x + 1 x 1 () Find the f 1 (x) if y = x. (a) f 1 (x) = x () f 1 (x) = x (c) f 1 (x) = ± x (d) There is no inverse function ecause f(x) = x is not one-to-one. (5) Find lim x e x+1. (a) () (c) no limit (d) 0 (6) Find the average value of the function f(x) = e x + e x on [ 1, 1]. (a) The average does not exist. () 0 (c) e e 1 (d) e + e 1 (7) Find the volume of the solid otained y rotation around x axis of the region ounded y y-axis, the graph of y = e x, the line x = 1 and the x-axis. (a) π(e 2 + 1) () π 2 (e2 1) (c) π(e 2 ) (d) π 2 (e2 + ) 11

12 Key 1d, 2d, d, d, 5d, 6, 7d, 8, 9d, 10, 11c, 12, 1c, 1d, 15d, 16e, 17c, 18, 19e, 20a 21 22, 2a, 2c, 25c, 26, 27e, 28a, 29a, 0d 1d, 2c, c, d, 5d, 6c, 7. 12