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Unformatted text preview: Name: Calculus I; Fall 2007 Part I
Part I consists of 6 questions, each worth 5 points. Clearly show your work for each of the problems listed. Evaluate the following limits: (1) 2x5  3x2 + 7 lim x 3x5 + 3x3  100 (2)
x5 lim sin(x + 2) x2 + 7 (3) x2  2x + 1 x1 x2  1 lim 1 2 (4) Given that 1 x sin(x) x2 +x 1 x for x > 0, find limx sin(x) x2 +x (5) If f (3) = 5 and f (3) = 1, find the equation of the tangent line to the graph of y = f (x) at x = 3. (6) Given the graph of the function y = f (x) below, list all places where this function is not continuous. 3 Part II
Part II consists of 5 problems each worth 14 points. If a problem has two parts, the first is worth 10 points and the second 4 points. Displaying only answers (even if correct) will not get you any points. You must show the relevant steps and justify your answer to earn credit. 1 (1) (a) If f (x) = x , find f (3) using the limit definition. (b) Find the equation of the tangent line to f (x) = 1 x at x = 3. 4 (2) Evaluate the limit: limx x x + 100 (3) If S(t) = t2 + 2t gives the position of a ball (in meters above the ground) at time t seconds (0 t 2). (a) Find the instantaneous velocity (using the limit definition) at time t = 1. (b) What can you say about the position of the particle at time t = 1? 5 (4) Given the function f (x) = x2 + 2 x + 8 when x 2, when x < 2. (a) Is the function y = f (x) continuous at x = 2? (b) Is the function y = f (x) differentiable at x = 2? As always you must explain your answers! 6 (5) Given the graph of the function below, draw a reasonable graph of its derivative. [I suggest that the graph is a horizontal and 1 vertical shift of y = x ] ...
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 Fall '07
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 Calculus, Derivative, Limits, lim, Continuous function

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