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Name: Date: Chapter 3 Test Review Calculator Not Allowed Directions: Find all critical points for the following functions. Determine where each of the following functions is increasing and decreasing. Classify all maximum and minimum values. 1. f ( x) x 4 3x 2 2 2. f ( x) 2 x3 3x 2 12 x 5 3. f ( x) ( x 3)2 4. f ( x) x 5. f ( x) 1 x x2 8 x 1 Directions: Determine where each of the following functions is concave up and down. Find all inflection points. 6. f ( x) x3 3x3 4 x 1 7. f ( x) x 4 6 x 2 2 x 3 Page 1 Directions: Given the following derivatives, find the local extrema and inflection points for f(x). 8. f '( x) x 4 16 9. f '( x) ( x 5) 2 Directions: Determine whether the Mean Value Theorem applies on the given intervals. If they do, find the exact value of all c that satisfy the conclusion of the MVT. 10. f ( x) x 2 on [-2,1] 11. f ( x) x 2 on [0,3] 12. f ( x) x3 x 2 2 x on [-1,1] 13. f ( x) ( x 3)( x 1) on [-1,0] Page 2 Directions: Sketch a graph of each of the following functions. Find the zeros, y-intercept, max/min, increasing, and decreasing. 14. f ( x) x3 ( x 4) -10 15. f ( x) ( x 2)( x 1) 6 6 4 4 2 2 -5 5 10 -10 -5 5 -2 -2 -4 -4 -6 -6 10 Calculators Allowed Directions: Sketch a graph of a function with the following characteristics. 16. x f'(x) 17. ( , 2) ( 2, 4) (4, ) Positive Negative Positive ( , 3) x f(x) Negative Positive 6 6 4 4 2 2 -5 5 10 -10 -5 5 -2 -2 -4 -4 -6 -6 Page 3 (-1,3) (3, ) Negative Positive (-2,1) (1, ) Negative Positive ( , 2) x f’(x) -10 (-3,-1) Positive 10 18. A 10 ft ladder leans against the side of a building. If the top of the ladder begins to slide down the wall at a rate of 2 ft/s, how fast is the bottom of the ladder sliding away from the wall when the ladder is 8 feet from the ground? 19. Oil spills out of a tanker at a rate of 120g/m. The oil spreads in a circle. Determine the rate at which the radius of the spill is increasing when the radius is 100ft. 20. Sand is dumped such that the shape of the sand pile remains a cone with height equal to the radius. If the sand is dumped at the constant rate of 20 cubic ft/s, find the rate at which the height is increasing when the height is 6 ft. 21. You have 1200ft of fencing. What is the largest area you can create? 22. What size squares should be cut out if you have a piece of material that is 15 by 11 and want to create an open-top box? Page 4