# exam1 - Basic Calculus/Exam#1/Spring 2012 NAME Calculators...

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Basic Calculus/Exam #1/Spring 2012 NAME: Calculators may be used. For full credit, you must show your work. You do not need to simplify your answers. Answer all questions. Total points: 100. 1. (10 points) Let f ( x ) = x 2 - 4 x + 2, g ( x ) = ( x 2 + 1) 2 . Find (a) f (1) (b) g (1) (c) g ( a - b ) (d) f ( g (1)) (e) g ( f ( x )) 2. (5 points) Find the equation of the straight line joining the points (1 , 2) and (9 , 6). 3. (5 points) Find the points of intersection of the curve y = x 2 + 6 and the line y = 7 x - 6.

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2 4. (10 points) Use the definition of derivative as a limit to find f 0 ( x ) if f ( x ) = x 2 - 6 x + 5 . 5. (5 points) Find the equation of the tangent line to the graph of f ( x ) = x 2 - 6 x + 5 at the point where x = 2. 6. (25 points) Use rules for differentiation to find the derivatives of the fol- lowing. (a) f ( x ) = x 3 + 2 x 2 + 2 x - 1 (b) f ( x ) = 1 x + 1 x (c) f ( x ) = ( x 4 + 2) 9 / 2 (d) f ( x ) = x ( x + 2)( x - 2) (e) f ( x ) = (( x 2 + 3 x + 4) 3 + 5) 4
3 7. (15 points) Let f ( x ) = 2 x 3 - 3 x 2 . (a) Find f 0 ( x ) and f 00 ( x ). (b) Find the region on which f is increasing. (c) Find the region on which f is concave down. (d) Sketch the graph of f , giving the coordinates of any relative maxima

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Unformatted text preview: and minima or inﬂection points. 8. (5 points) Find the positive numbers x and y that satisfy xy = 100 and which minimize 2 x + y . 4 9. (10 points) A ball is thown up in the air at a speed of 96 feet per second from a point 48 feet above the ground. The height in feet s ( t ) of the ball at time t seconds is given by s ( t ) = 32 + 64 t-16 t 2 . (a) When does the ball reach its maximum height? (b) What is the ball’s maximum height? (c) When does the ball hit the ground? (d) How fast is the ball going when it hits the ground? 10. (10 points) When x items are produced, the cost is C ( x ) = 100 x +10 , 000 and the price (demand curve) is p ( x ) = 400-. 01 x . (a) What is the revenue R ( x )? (b) What is the proﬁt P ( x )? (c) Find the production level x that maximizes proﬁt, and the maximum value of proﬁt....
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• Spring '12
• randall

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