t1(1) - v ( t ). (b) Give the acceleration function a ( t...

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Methods of Calculus MAC3233 Test 1 Feb. 1, 1993 Instructions: Do not write your answers on this paper, use separate answer sheets. This is a closed-book test. No calculators are allowed to be used. (1) (12 pts.) f ( x ) = 1 x 3 . (a) f 0 ( x ) = ? (b) f 0 (2) = ? (2) (8 points) d dx x 3 / 4 · = ? (3) (10 points) f ( x ) = 3 p 5 x 4 + x . f 0 ( x ) = ? (4) (10 points) f ( x ) = x 2 + 2 /x . Find the equation of the tangent line at (2 , 5). (5) (10 points) The daily output of a factory assembly line is about 60 t + t 2 - 1 12 t 3 units produced after t hours of work, where 0 t 8. What is the instantaneous rate of production when t = 4? (6) (15 points) An object travels in a straight line. The distance travelled in t hours is s ( t ) = t 2 + 3 t kilometers. (a) Give the velocity function
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Unformatted text preview: v ( t ). (b) Give the acceleration function a ( t ). (c) What is the initial velocity? (d) What is the velocity when t = 5 hours? (e) What is the acceleration when t = 5 hours? (7) (10 points) Sketch the graph of a function satisfying: f (-2) = 4, f (4) = 0, f (-2) = 0, f (4) = 0, f 00 ( x ) < 0 for x < 1, and f 00 ( x ) > 0 for x > 1. (8) (25 points) For the funtion f ( x ) = 11 + 9 x-3 x 3-x 3 . (a) Give f ( x ). (b) Give f 00 ( x ). (c) Find all extrema. Use f 00 ( x ) to say whether each is a relative maximum or minimum. (d) Find all inection points. (e) Sketch the graph. 1...
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