MATH151 Fall 2022 Common Exam II Version A FIRST NAME print UIN LAST NAME print INSTRUCTOR SECTION NUMBER DIRECTIONS No calculators cell phones smart watches headphones or other electronic devices may be used and must be put away TURN OFF cell phones and put them away If a cell phone is seen during the exam your exam will be collected and you will receive a zero In Part I mark the correct choice on your ScanTron using a No 2 pencil The scantrons will not be returned therefore for your own records also record your choices on your exam In Part II present your solutions in the space provided Show all your work neatly and concisely and clearly indicate 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 Be sure to fill in your name UIN section number and version letter of the exam on the ScanTron form THE AGGIE HONOR CODE An Aggie does not lie cheat or steal or tolerate those who do Signature 1 1 7e6 Part I Multiple Choice 3 5 points each 6 t te3t at the point where 1 Find the slope of the tangent line to the curve 3 y xy2 18 at the point 3 2 22 a 154 b 922 c 98 d 34 e 15 2 Find a tangent vector to the curve r t t 2 a 1 b 3e6 c 1 4444 7e6 d 1 3e6 1 e6 e 3 Find the 123rd derivative of f x 1 cos 3x a 3 122 cos 3x b 3122 sin 3x c 3 122 sin 3x d 3123 cos 3x e 3 123 sin 3x 3 2 2 2 4 There are two lines tangent to the parabola f x x2 1 that pass through the point 1 2 Find the x coordinates where these tangent lines touch the parabola a x 1 3 b x 0 2 c x 2 4 d x 1 5 e None of these 5 Which of the following is true 3x2 4 if 1 x 3 2x3 5 if x f x 10x 1 if x 3 1 a f is continuous but not differentiable at x 1 f is differentiable at x 3 b f is continuous but not differentiable at both x 1 and x 3 c f is differentiable at both x 1 and x 3 d f is differentiable at x 1 f is continuous but not differentiable at x 3 e f is discontinuous at both x 1 and x 3 3 6 Find the slope of the tangent line to the curve x 6t2 26 y t2 3t 2 at the point 2 0 7 a 241 b 241 c 2424 d 7 e 24 7 Find the linearization of f x x3 2 at a 4 and use it to approximate f 4 1 a 7 9 b 8 1 c 8 2 d 8 3 e 8 4 8 Find the t value s where the curve x sin2 t cos t y t cos t 0 t 2 has a vertical tangent line a 5 3 3 5 b 0 3 3 11 c 0 6 63 d 2 3 5 e 0 2 3 3 4 3 t2 5t where s is in feet 9 The position of a particle moving in a straight line is given by s t 2 t3 and t is in seconds Find the velocity at the time where the acceleration of the particle is 10 ft s2 a 3 ft s b 9 ft s c 17 ft s d 24 ft s e 5 ft s 10 Find the slope of the tangent line to f x arctan 2x at x 3 2 a 372 b 191 c 371 d 191 e 7 11 A bacteria culture grows at a rate proportional to its size It has 100 bacteria initially and 400 bacteria after 3 hours How long will it take for there to be 1000 bacteria 4 ln 10 ln 3 hours a b ln 10 3 ln 4 hours 30 c ln 4 hours 3ln 10 d ln 4 hours e None of these 5 12 An object is in motion according to s t t2 4t 7 t 0 where t is in seconds and s is in feet Find the total distance traveled by the object during the first three seconds a 2 ft b 3 ft c 4 ft d 5 ft e 7 ft 13 The radius of a sphere was measured to be 10 cm with a maximum error in measurement of 0 2 cm Use differentials to estimate the maximum error in the calculated surface area of the sphere The surface area of a sphere is S 4 r2 a 4 cm2 b 16 cm2 c 8 cm2 d 40 cm2 e 80 cm2 14 Find the derivative of f x ln 1 a 3 csc x 8x34x3 b 3 cot x x4 5 2x3 c 3 cot x 2x3 3 x4 5 x4 5 d sec x tan x 4x3 e 3 csc x cot x x4 5 csc3 x x4 5 2 6 x g x 4 2 Suppose f and g are differentiable functions Use the following table of values to answer questions 15 and 16 f x g x 15 Find H 1 if H x g f x f x 1 2 3 1 1 4 5 1 2x 1 20 a 94 b 364 c 92 d 94 e 9 16 Find h 1 if h x f x2 2 a 4 b 8 c 24 d 16 e Not enough information 7 Part II Work Out Problems 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 17 15 points Differentiate the following functions Do not simplify after taking the derivative a y sec3 3x5 7 2x tan 4x b y x3 e 2 c y log10 5 x2 e x 8 18 8 points Differentiate y x5 4x3 arcsin 2x Your answer should be in terms of x only 19 9 points Find dy dxfor sin xy2 ey x 9 9 20 12 points A camera is positioned 8 m away from the launching point of a weather balloon The balloon is released and rises vertically at a speed of 3 m s a At what rate is the distance between the camera and the balloon changing when the balloon has risen 6 m b If the camera is always kept aimed at the balloon how fast is the camera s angle of elevation changing when the balloon has risen 6 m 10 Do not write in this table Question Points Awarded Points 1 16 56 15 17 18 8 9 19 20 12 100 Total 11
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