MATH 140A MIDTERM EXAMINATION II March 27, 2003Name ID # Section #There are 10 multiple choice questions, 5 True/False questions, and 4 partial credit questions.THE USE OF CALCULATORS IS NOT PERMITTED IN THISEXAMINATION.There are 15 problems on 11 pages, including this one.Check your booklet now.The box below is for the instructor’s use.MC ....................(50)T/F ...................(10)12 ....................(6)13 ....................(10)14 ....................(12)15.....................(12)Total ............... ( )MATH 140A MIDTERM EXAMINATION II PAGE 21. (5 pts.) The slope of the normal line, at the point (−1, 1), of the graph given by x2+ y2= 2isa) −1b) 0c) 1d) 2e) The normal line does not exist at the point.2. (5 pts.) Suppose x = sin y, finddydx.a) 0b)1xc)1√1 − x2d)11 + x2e) x2− 1MATH 140A MIDTERM EXAMINATION II PAGE 33. (5 pts.) Suppose f (x) = x +1x. Find all critical points of f (x).a) x = 0b) x = 1c) x = 0, 1d) x = 0, 1, −1e) x = 1, −14. (5 pts.) Suppose f0(x) = x(x − 2)2(x − 4). How many local maximum does f (x) have?a) 0b) 1c) 2d) 3e) 4MATH 140A MIDTERM EXAMINATION II PAGE 45. (5 pts.) Find the value of x = c that satisfies the conclusion of the Mean Value Theorem forthe function f(x) =√2x − 1 on the interval [5, 13].a) 4b)112c) 6d)172e) 126. (5 pts.) Suppose f00(x) =x2− 4x. Find all points of inflection of y = f(x).a) x = 0b) x = 2c) x = 0, 2d) x = 2, −2e) x = 0, 2, −2MATH 140A MIDTERM EXAMINATION II PAGE 57. (5 pts.) If a, b, c, d, e, f, and g are positive constants, then limx→∞2(ax + b)(cx + d)ex2+ fx + g=a) 0b)2ec)2aed)2acee) ∞8. (5 pts.) What are the horizontal (H.A.) and the vertical (V.A.) asymptotes of the functionf(x) =x + 1x2− 2x − 3?a) H.A. y = 0, V.A. x = 3b) H.A. y = 1, V.A. x = 3c) H.A. y = 0, V.A. x = −1 and x = 3d) no H.A., V.A. x = −1 and x = 3e) There are no asymptotes.MATH 140A MIDTERM EXAMINATION II PAGE 69. (5 pts.) Suppose x and y are 2 positive numbers whose sum is 10. What is the maximumvalue of the product x2y?a) 0b)400027c)2009d) 125e) 2510. (5 pts.) Suppose f (1) = 3 and f0(1) = −1. Then the linear approximation L(1.5) of f (x) atx = 1.5 isa) 1.5b) 2c) 2.5d) 3e) 3.5MATH 140A MIDTERM EXAMINATION II PAGE 711. (10 pts.) True or Falsea) The function f (x) = x3+ 3x + 10 has no critical points. T Fb) If f0(c) does not exist, then x = c cannot be a local maximum or a local minimumpoint of f(x). T Fc) If two functions have the same dvivative, then they must be the same function. T Fd) Any differentiable function f (x) defined on the interval (a, b) have (at least) oneabsolute maximum and one absolute minimum on (a, b). T Fe) If f00(c) = 0, then x = c might be a point of inflection of y = f(x). T FMATH 140A MIDTERM EXAMINATION II PAGE 812. (6 pts.) On an interval I, suppose f (x) and g(x) are two differentable functions that areboth negative, increasing, and concave down. Then on I, their product f(x)g(x) must be(choose 1 each)a) Positive Negativeb) Increasing Decreasingc) Concave up concave downMATH 140A MIDTERM EXAMINATION II PAGE 913. (10 pts.) Car A is traveling west at 30 miles per hour and car B is traveling north at 50miles per hour. Both cars are headed for the intersection of the two roads. At what rate isthe distance between them changing when car A is 4 miles and car B is 3 miles from theintersection?MATH 140A MIDTERM EXAMINATION II PAGE 1014. (12 pts.) For the function f(x) = x3− 6x2+ 3a) Find all critical points of f(x).b) Determine if each critical point is a local maximum or a local minimum.c) Find the absolute maximum and the absolute minimum points of f(x) on the interval[−1, 3].MATH 140A MIDTERM EXAMINATION II PAGE 1115. (12 pts.) A can in the shape of a circular cylinder has a volume of 32π cubic inches. Thematerial used for the top and bottom of the can costs twice as much as the material usedfor the side of the can. Find the radius of the can that minimizes the cost of the