MATH 140 A EXAMINATION II, FORM A SPRING 20061. If y = 2x3+ 4x anddxdt= 5, finddydtwhen x = 1.a) 10b) 50c) 100d) 200e) 2702. Two cars start moving from the same point. One travels south at30 mi/h and the other travels east at 40 mi/h. At what rate is thedistance between the cars increasing 1 hour later?a) 60 mi/hb) 55 mi/hc) 42 mi/hd) 45 mi/he) 50 mi/h3. Find the linearization L(x) of f (x) =1√3 + xat a = 0.a) L(x) = −1√3+x6√3b) L(x) =√3 − 6√3xc) L(x) =1√3+x6√3d) L(x) =1√3−x6√3e) L(x) =√3 + x4. Find the differential of the function y = x4+ 2x.a) dy = (x3+ 2)dxb) dy = (4x4+ 2)dxc) dy = (4x − 2)dxd) dy = (4x3+ 2)dxe) dy = (x5+ x2)dx5. Find the critical number of the function y = 2x2+ 20x.a) 2b) 0c) −5d) 20e) −106. Find the absolute minimum value of y = 5x2−30x+4 on the interval[0, 4].a) 3b) −41c) 30d) −45e) −307. Find the absolute maximum value of y =p4 − x2on the interval[−2, 2].a) 1b) 2c) 3d) 0e) 48. Find the exact values of the numbers c that satisfy the conclusionof The Mean Value Theorem for the function f(x) = x3−4x for theinterval [−4, 4].a) c = ±4√3b) c = ±2c) c = ±√3d) c = ±4√33e) c = 0, 49. How many real roots does the equation x5+ 9x + 3 = 0 have?a) Exactly one real root.b) Exactly two real roots.c) No real roots.d) Exactly three real roots.e) Exactly five real roots.10. The graph of the first derivative f0(x) of a function f (x) is shownbelow. At what value(s) of x does f (x) have a local maximum?a) 3 and 8b) 5 and 10c) 6d) no loca l maximume) 2 and 71MATH 140 A EXAMINATION II, FORM A SPRING 200611. Find the intervals of the increase or decrease of the function f (x) =x3− 15x2+ 63x.a) f is decreasing on (−∞, 3) and (7, ∞), and increasing on(3, 7)b) f is increasing on (−∞, 9) and (21, ∞), and decreasing on(9, 21)c) f is decreasing on (−∞, 9) and (21, ∞), and increasing on(9, 21)d) f is increasing on (−∞, 3) and (7, ∞), and decreasing on(3, 7)e) f is increasing on (−∞, 0) and decreasing on (0, ∞)12. How many points of inflection are on the graph of the functionf(x) = −15x3+ 4x2− 8x − 20?a) 2b) 4c) 1d) 3e) 013. Evaluate limx→∞x + 4x2− 2x + 7.a) 1b) ∞c)47d) 0e) 414. Evaluate limx→∞√x2+ 6x6x + 7.a)17b) ∞c) 0d)76e)1615. Find the horizontal asymptote of the curve y =1 − 6x1 + x.a) y = 0b) y = 6c) y = −6d) y = 1e) y = −116. Find the point on the line y = 2x + 7 that is closest to the origin.a)„−135,225«b) (−7, −7)c) (7, 21)d)„−165,195«e)„−145,75«17. Which one of the following statements is correct for the graph ofy = 8x2− x4?a) The graph is increasing on (−∞, 0).b) The graph is increasing on (2, ∞).c) The graph is decreasing on (−∞, −2).d) The graph is increasing on (−2, 0).e) The graph is increasing on (0, 2).18. Which one of the following statements is correct for the graph ofy = 8x2− x4?a) The graph is concave downward on the interval −∞, −2√33!.b) The graph is concave upward on the interval (0, ∞).c) The graph is concave upward on the interval (−∞, 0).d) The graph is concave downward on the interval −2√33,2√33!.e) The graph is concave upward on the interval (−2, 2).2MATH 140 A EXAMINATION II, FORM A SPRING 200619. Find the difference of two positive numbers whose product is 144and whose sum is a minimum.a) 0b) 12c) 24d) 6e) 820. A rectangular storage container with an open top is to have a volumeof 10 m3. The leng th of its base is twice the width. Material for thebase costs $10 per square meter. Material for the side costs $6 persquare meter. Find the width for the cheapest such container.a)3r12mb)3r29mc)r32md)r92me)3r92m21. (10 pts.) If a snowball melts so that its surface area decreases at arate of 1 cm2/min, find the rate (in cm/min) at which the diameterdecreases when the diameter is 27 cm . (Hint: Surface Area = 4πr2=πd2).22. (10 pts.) Find the absolute minimum value of y = 4x2+8xon theinterval»12, 8–.23. (10 pts.) Find the largest possible volume of a box with a squarebase and an open top whose total surface is 1200 cm2.24. (20 pts.) Consider the function f (x) =xx2− 9where f0(x) =−(x2+ 9)(x2− 9)2andf00(x) =2x(x2+ 27)(x2− 9)3.(a)(2 pts.) Find the domain of f (x).(b)(1 pt.) Find the x- and y-intercepts of the graph of f (x).(c)(1 pt.) Is the graph of f(x) symmetric ab out the y-axis, symmet-ric ab out the origin or neither?(d)(3 pts.) Find the horizontal, vertical and slant asymptotes forf(x) if any exist.(e)(3 pts.) Find all the critical numbers for f (x).(f)(2 pts.) Find the interval(s) where f(x) is increasing and theinterval(s) where f (x) is decreasing.(g)(2 pts.) Find the local minimum and local maximum po ints off(x).(h)(2 pts.) Find the interval(s) where f(x) is concave up and theinterval(s) where f (x) is concave down.(i)(2 pts.) Find all the inflection points for f (x).(j)(2 pts.) Sketch the graph of f