Math150Sols3S11

Math150Sols3S11 - QUIZ ON CHAPTER 3 - SOLUTIONS MATH 150...

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QUIZ ON CHAPTER 3 - SOLUTIONS MATH 150 – SPRING 2011 – KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% Show all work, simplify as appropriate, and use “good form and procedure” (as in class). Box in your final answers! No notes or books allowed. A scientific calculator is allowed. 1) Use the limit definition of the derivative to prove that D x cos x () = ± sin x , as we have done in class. Show all steps! (12 points) NOTE: THIS IS THE ONLY PROBLEM ON THIS TEST WHEN YOU WILL NEED TO USE THE LIMIT DEFINITION OF THE DERIVATIVE! Let fx = cos x . ± = lim h ² 0 + h ³ h = lim h ² 0 cos x + h ³ cos x h = lim h ² 0 cos x cos h ³ sin x sin h from Sum ID for cosine ±² ³³³ ³´ ³³³³ ³ cos x h = lim h ² 0 cos x cos h ³ cos x Group terms with cos x . ³³ ³ sin x sin h h = lim h ² 0 cos x cos h ³ 1 ³ sin x sin h h = lim h ² 0 cos x cos h ³ 1 h ´ µ · ¸ ¹ ² 0 µ¶ ³· ³ ³ sin x sin h h ´ µ · ¸ ¹ ² 1 ³ º » ¼ ¼ ¼ ¼ ½ ¾ ¿ ¿ ¿ ¿ Group " h stuff" = ³ sin x Q.E.D.
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2) A particle moving along a coordinate line has as its position function s , where st () = 3 t + 2 2 3 , for t ± 1 . Position s is measured in meters, and time t is measured in seconds. Find the velocity of the particle at time t = 8 (seconds). Write an exact answer using correct units. (8 points) = 3 t + 2 t 2 3 = 3 t + 2 t ± 2/3 ² vt = ³ = 3 + 2 ± 2 3 t ± 5/3 ´ µ · ¸ ¹ = 3 ± 4 3 t ± = 3 ± 4 3 t ² v 8 = 3 ± 4 38 = 3 ± 4 3 5 = 3 ± 4 32 5 = 3 ± 4 96 = 3 ± 1 24 = 2 23 24 meters second or 71 24 meters second
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3) If fx () = 9 ± x 2 , is f differentiable on the interval ± 3, 3 ² ³ ´ µ ? Box in one: (2 points) Yes No Observe that the graph of f below has (one-sided) vertical tangent lines at x 3. Also, observe that ± is undefined at x 3: = 9 ± x 2 = 9 ± x 2 1/2 ² ³ = 9 ± x 2 ± ± x = ± x 9 ± x 2
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4) Let fx () = 3 x ± 1 3 . Find an equation of the tangent line to the graph of f at the point 1, 8 . You may use any form. (8 points) Find the slope, m , of the tangent line.
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Math150Sols3S11 - QUIZ ON CHAPTER 3 - SOLUTIONS MATH 150...

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