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Unformatted text preview: Calculus Group I (1) Let f ( x ) = 1 π arctan ( cot ( π x )) + x- 1 2 . The domain of f is R \ Z . (a) Let x ∈ R \ Z . Show that f ( x ) = 0. (b) Let a ∈ ( n , n + 1 ) . Find the exact value of f ( a ) in terms of n . You may use the result from part (a). (c) The function f agrees with a great function in mathematics on R \ Z . What is that function? Explain. Calculus Group I (2) Find a cubic polynomial whose graph contains the points (0,-1) and (1,1) such that the tangent line to the graph at the point (0,-1) has slope 3 and the point (1,1) is an inflection point of the graph. Calculus Group I (3) The graph of the hyperbola y 2- x 2 = 1 has a pair of parallel tangent lines with positive slope that are a distance of 1 apart. Find the equations of these lines. Hint : the distance d between the line Ax + By + C = 0 and the point ( x 1 , y 1 ) is given by the formula d = | Ax 1 + By 1 + C | √ A 2 + B 2 Calculus Group I (4) At the ABC Peanut Butter Plant, the rate of production of peanut butter is mea- sured in gallons per hour during the production shift from 6am to 6pm, and is given by the formula p ( t ) = 336 t- 0.781 t 3 + 1430.7- 7729.4 t- 2 , where t = 0 corresponds to midnight. (a) Find the total amount of peanut butter produced between 10am and 11am. Round your answer to the nearest hundredth. (b) Find the average rate of production during the 6am to 6pm shift. Round your answer to the nearest hundredth. (c) When does the maximum production rate occur during the 6am to 6pm shift? Round your answer to the nearest minute. (d) When is the rate of production increasing the most? Round your answer to the nearest minute....
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This note was uploaded on 09/26/2011 for the course ECE 123 taught by Professor Crank during the Spring '11 term at LSU.
- Spring '11