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1 E3-151-09b-copyright Joe Kahlig 1. (18 points) Find dy for the following. dx (a) y = (x5 + 7)x (b) y = log3 (x2 + 2) + arctan(5x) (c) y = sin−1 (x2 ) + 5x 3 2. (6 points) You have been given a sample of a chemical that has a half life of 4 months. How long will it take until 20% is gone? E3-151-09b-copyright Joe Kahlig 3. (9 points) Simplify. When possible, give exact value. 7π = (a) arcsin sin 5 7π = (b) arccos cos 5 x = (c) sin cos−1 3 (x − 4)2 (9 − x) . (x − 1)3 Find the interval(s) where f (x) is increasing and the intervals where f (x) is decreasing. 4. (6 points) The domain of f (x) is all real numbers except x = 1. Use f ′ (x) = 2 3 E3-151-09b-copyright Joe Kahlig 5. (10 points) Compute these limits. Give exact values of these limits. x 1 (a) lim − x − 1 ln(x) x→1+ −1/x2 (b) lim (cos (3x)) x→0 4 E3-151-09b-copyright Joe Kahlig 6. (6 points) Find the absolute maximum and the absolute minimum of f (x) on the interval [−2, 0] when 2 f (x) = 50xe−x 7. (12 points) Below is the graph of the derivative f ′ (x) of a function f (x). Assume that the function f (x) is continuous for all real numbers. Use the graph to answer the following questions. 5 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 1 2 3 4 5 6 −2 −3 −4 (a) Classify the critical values of f (x) as local max, local min, or neither. (b) Find the interval(s) where f (x) is both increasing and concave down. (c) Give the x-values of all inflection points. 5 E3-151-09b-copyright Joe Kahlig 8. (6 points) Find the constants a and b such that the point (1, −4) is an inflection point for f (x) = ax3 + bx2 + 10. 9. (10 points) Find f (x). √ 3 x2 − 7 ′ (a) f (x) = x (b) f ′ (x) = 30 + 2sec2 (x) e3x E3-151-09b-copyright Joe Kahlig 6 10. (7 points) Find f (x) if f (1) = 10 and f ′ (x) = 1 +9 5x2 11. (10 points) A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions for the poster will give the largest printed area? Be sure to show that your answer satisfies the above conditions.