F x 8 x x if x if x 0 x cos x if x if you think f is

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f ( x ) = 8 < : x | x | if x < 0 0 if x = 0 x - cos x if x > 0 If you think f is not continuous at x = 0, what kind of discontinuity is it? 5. (10 points) Calculate the equation of the tangent line to the curve defined by x 3 + y 3 = 2 xy at the point, (1 , 1). 6. Let f ( x ) = 4 + 8 x 3 - x 4 (a) (2 points) Over what x -intervals is f increasing/decreasing? (b) (2 points) Find all critical points of f and test each for local maximum and local minimum. (c) (3 points) Over what x -intervals is f concave up/down? (d) (3 points) Sketch the shape of the graph of f . 7. Compute (a) (3 points)lim x ! 0 x 3 - p 9+ x (b) (3 points) lim x ! sin x - x (c) (4 points) lim x !- 2 x 2 - x - 6 x 3 +8 2
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8. Microtutorial (10 points) If W denotes the weight in pounds of an individual, and t denotes the time in months, then dW/dt is the rate of weight gain or loss in lbs/mo. The current speed record for weight loss is a drop in weight from 487 pounds to 130 pounds over an eight-month period. Show that the rate of weight loss exceeded 44 lbs/mo at some time during the eight-month period. 9. Let g ( x ) = ( 2 x 2 - 8 x ) 2 3 (a) (5 points) Find all critical points of g over the x -interval [0 , 8]. (b) (5 points) Find absolute maximum and absolute minimum of g over [0 , 8]. 10. Let y = tan ( x ) (a) (5 points) Find the di erential dy of y = tan ( x ) at x = 4 . (b) (5 points) Use di erentials to find an approximate value for tan (0 . 885). Hint: 4 0 . 785. 3
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  • Winter '07
  • APOORVA
  • Calculus, Derivative, lim, Mathematical analysis, Continuous function, Department of Mathematics

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