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Calculus I Exam #1 Fall 2005 Name _____________________________ Show all your work on this paper. Solutions without correct supporting work will not be accepted. All solutions must be exact unless stated otherwise in the problem. 1. Use the graph of y f (x ) given below to find the limits indicated. (2 points each) a. lim f ( x) _______________ x2 b. lim f ( x) _______________ x 1 c. lim f ( x) _______________ x 4 2. Evaluate without using a graph or table. (i.e., show that you worked it analytically) a. lim x 5 c. lim 0 x 2 x 3 = ________________ cos tan = ________________ (5 pts each) 3x 3 3x = ________________________ x 1 2 x 2 2 x b. lim d. lim x2 1 x = ____________________________ 2 x 3. Find all points of discontinuity. Also state which part of the definition of continuity fails at that particular point. If there are none make sure you write ‘NONE” in the blank. (3 points each) x 1 a. f ( x) Discontinuous at x = _________ Reason: ________________________ x3 Removable or Nonremovable? ________________________ 4 x , x 5 b. f ( x) 2 x 11, x 5 Discontinuous at x = _________ Reason:__________________________ Removable or Nonremovable? ________________________ 4. Find f (x) using the definition of the derivative: f ( x) 3 x 2 x 5 5. Find the derivative. Do not simplify your answers. Circle your answers. a. y 4x3 3 x5 b. f ( x) ln(sin x) c. g ( x) d. x 5 9e x x2 3 y x 3 4x 2 9 7 sec x x (8 points) (7 points each) 6. Write the equation of the tangent line to the graph of f ( x) arctan x at the point where x 1. Write your answer in slope-intercept form. (8 points) 7. Find dy : x 3 y 2 y 4x 5 dx (8 points) 8. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building (in meters) if the splash is seen 6.8 seconds after the stone is dropped? Recall that the position function is given by s (t ) 4.9t 2 v0 t s 0 . (8 points) 9. Find dy : dx y ( x 3) x (8 points)