MATH 151, FALL 2011COMMON EXAM II - VERSION BLast Name: First Name:Signature: Section No:PART I: Multiple Choice (15 questions, 4 points each. No Calculators)Write your name, section number, and version letter (B) of the exam on the ScanTron form.1. Find the derivative of g (x) =x3+ 1x2+ 1.(a) g′(x) =x4+ 3x2− 2x(x2+ 1)2(b) g′(x) =5x4+ 3x2+ 2xx2+ 1(c) g′(x) =x4+ 3x2+ 2x(x2+ 1)2(d) g′(x) =x4+ 3x2− 2xx2+ 1(e) g′(x) =5x4+ 3x2+ 2x(x2+ 1)22. A ball is thrown vertically upward with a velocity of 80 feet per second and the height, s, of the ball at time tseconds is given by s(t) = 80t − 16t2. What is the velocity of the ball when it is 96 feet above the ground on itsway up?(a) 112 ft/sec(b) 24 ft/sec(c) 16 ft/sec(d) 48 ft/sec(e) 64 ft/sec3. Which of the following vectors is tangent to the curve r(t) =√t2+ 1, tat the point (2,√3)?(a)1√5, 1(b)12, 1(c)*√34, 1+(d)2√5, 1(e)*√32, 1+14. Find the 81stderivative of f (x) =1x.(a) f(81)(x) = −(81)!x81(b) f(81)(x) =(80)!x80(c) f(81)(x) = −(81)!x82(d) f(81)(x) = −(80)!x80(e) f(81)(x) =(81)!x815. limx→−∞(9 − 7e−x) =(a) −∞(b) 0(c) ∞(d) 7(e) 96. At what point on the graph of f (x) =√x is the tangent line parallel to the line 2x − 3y = 4?(a)169,43(b)916, 0(c)43,2√3(d)916,34(e)116,147. Given the equation 2xy + π sin(y) = 2π, finddydxwhen x = 1 and y =π2.(a) −π2 − π(b) −π3(c) −π2 + π(d) 0(e) −π228. Find the equatio n o f the tangent line to the graph of f (x) =x1 + 2xat x = 1.(a) y −13= −19(x − 1)(b) y −13= −49(x − 1)(c) y −13=x1 + 2x(x − 1)(d) y −13=19(x − 1)(e) y −13= −x1 + 2x(x − 1)9. If f (x) = sin(g(x)), find f′(2) given that g(2) =π3and g′(2) =π4.(a)π8(b)√3π8(c)12(d) −√3π8(e) −π810. limx→0sin3(4x)x3=(a) ∞(b) 64(c) 1(d) 0(e) 411. Find the slope o f the tangent line to the curve x = t2+ t + 1, y =√t + 4 at t = 9.(a)1114(b)35(c)196(d)512(e) 114312. Find the der ivative of h(t) = (t4+ 7t)5.(a) h′(t) = 5(4t3+ 7)4(b) h′(t) = 5(t4+ 7t)4(4t3)(c) h′(t) = 5(t4+ 7t)(4t3+ 7)(d) h′(t) = 20t19+ 75(5t4)(e) h′(t) = 5(t4+ 7t)4(4t3+ 7)13. Given f (x) is a one-to-one function, find g′(3) where g is the inverse of the function f (x) = x9+ x3+ x.(a) g′(3) =112(b) g′(3) = 1(c) g′(3) =113(d) g′(3) =19(e) g′(3) = 1314. Find the der ivative of f(x) = x3e2x.(a) f′(x) = 3x2e2x+ 2x3e2x(b) f′(x) = 6x2e2x(c) f′(x) = 3x2e2x+ x3e2x(d) f′(x) = 3x2e2x(e) f′(x) = 3x2e2x+ 2x4e2x−115. Find the linear approximation, L(x), for f (x) =3√x at x = −8.(a) L(x) = −2 +112(x + 8)(b) L(x) = −2 −112(x + 8)(c) L(x) = −2 +112(x − 8)(d) L(x) = −2 −112(x − 8)(e) L(x) = −2 +14(x + 8)4PART II WORK OUTDirections: Present your solutions in the space provided. Show all your work neatly and concisely and Box yourfinal answer. You will be graded not merely on the final answer, but also on the quality and correctness of the workleading up to it.16. (8 pts) An observer is standing 8 feet from the base of a balloon launching point. At the instant the balloon hasrisen vertically 6 feet, the height of the balloon is increasing at a rate of 10 feet per minute. How fast is the distancefrom the observer to the balloon changing at this same instant? Assume the balloon starts on the ground and r isesvertically.17. (8 pts) Find the second derivative of f (x) = ta n(x3).518. (8 pts) A rain gaug e has the shape of a cone with the vertex at the bottom whos e radius is half of the height. Giventhat the volume of a cone is V =13πr2h, find the differential dV in terms of only h and the differential dh. Use thedifferential dV to estimate the change in volume when the height of water in the gauge increases from 5 cm to 5.3cm.19. (8 pts) For the equation y = e2x+ e−3x, show y′′+ y′− 6y is a co nstant. Find the constant.620. (8 pts) Draw a diagra m to show there are two tangent lines to the parabola y = 2 x2that pass thr ough the p oint(1, −3) by sketching the parabola and both tangent lines on the grid provided below. Find the x-coordinates wherethese tangent lines touch the parabola.7Last Name: First Name:Section No:Question Points Awarded Points1-15 6016 817 818 819 820
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