MATH 151 FALL 2011 COMMON EXAM II VERSION B Last 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 a g x x3 1 x2 1 x4 3x2 2x x2 1 2 5x4 3x2 2x x2 1 4 x 3x2 2x c g x x2 1 2 b g x x4 3x2 2x x2 1 5x4 3x2 2x e g x x2 1 2 d g x 2 A ball is thrown vertically upward with a velocity of 80 feet per second and the height s of the ball at time t seconds is given by s t 80t 16t2 What is the velocity of the ball when it is 96 feet above the ground on its way up a 112 ft sec b 24 ft sec c 16 ft sec d 48 ft sec e 64 ft sec 3 Which of the following vectors is tangent to the curve r t 1 a 1 5 1 b 1 2 3 1 c 4 2 d 1 5 3 1 e 2 1 t2 1 t at the point 2 3 4 Find the 81st derivative of f x 1 x 81 x81 80 f 81 x 80 x 81 f 81 x 82 x 80 81 f x 80 x 81 f 81 x 81 x a f 81 x b c d e 5 lim 9 7e x x a b 0 c d 7 e 9 6 At what point on the graph of f x 16 4 a 9 3 9 0 b 16 4 2 c 3 3 9 3 d 16 4 1 1 e 16 4 x is the tangent line parallel to the line 2x 3y 4 7 Given the equation 2xy sin y 2 find dy when x 1 and y dx 2 2 3 2 0 2 a b c d e 2 8 Find the equation of the tangent line to the graph of f x a y b y c y d y e y 1 3 1 3 1 3 1 3 1 3 1 x 1 9 4 x 1 9 x x 1 1 2x 1 x 1 9 x x 1 1 2x 9 If f x sin g x find f 2 given that g 2 a b c d e x at x 1 1 2x and g 2 3 4 8 3 8 1 2 3 8 8 sin3 4x x 0 x3 10 lim a b 64 c 1 d 0 e 4 11 Find the slope of the tangent line to the curve x t2 t 1 y a b c d e 1 114 3 5 19 6 5 12 114 3 t 4 at t 9 12 Find the derivative 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 1 12 g 3 1 1 g 3 13 1 g 3 9 g 3 13 a g 3 b c d e 14 Find the derivative of f x x3 e2x a f x 3x2 e2x 2x3 e2x b f x 6x2 e2x c f x 3x2 e2x x3 e2x d f x 3x2 e2x e f x 3x2 e2x 2x4 e2x 1 15 Find the linear approximation L x for f x a L x 2 b L x 2 c L x 2 d L x 2 e L x 2 3 x at x 8 1 x 8 12 1 x 8 12 1 x 8 12 1 x 8 12 1 x 8 4 4 PART II WORK OUT Directions Present your solutions in the space provided Show all your work neatly and concisely and Box your final answer You will be graded not merely on the final answer but also on the quality and correctness of the work leading 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 has risen vertically 6 feet the height of the balloon is increasing at a rate of 10 feet per minute How fast is the distance from the observer to the balloon changing at this same instant Assume the balloon starts on the ground and rises vertically 17 8 pts Find the second derivative of f x tan x3 5 18 8 pts A rain gauge has the shape of a cone with the vertex at the bottom whose radius is half of the height Given 1 that the volume of a cone is V r2 h find the differential dV in terms of only h and the differential dh Use the 3 differential dV to estimate the change in volume when the height of water in the gauge increases from 5 cm to 5 3 cm 19 8 pts For the equation y e2x e 3x show y y 6y is a constant Find the constant 6 20 8 pts Draw a diagram to show there are two tangent lines to the parabola y 2x2 that pass through the point 1 3 by sketching the parabola and both tangent lines on the grid provided below Find the x coordinates where these tangent lines touch the parabola 7 Last Name First Name Section No Question Points Awarded Points 1 15 60 16 8 17 8 18 8 19 8 20 8 100 8
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