Math 2211 4.1-4.8 Review Name________________________ Just Relax and Think Positively!!!! I Know you CAN Do This!!! 1. Find all critical numbers for the function . 2. Find all critical numbers for the function . 3. Find all critical numbers for the function . 4. Find all critical numbers for the function . 5. Find the value x at which the minimum of the function occurs. 6. Find the absolute maximum of the function on the interval . 7. Find the local extreme value(s), if any, of f(x) = x4 6x2. 8. Find all value(s) of c (if any) that satisfy the conclusion of the Mean Value Theorem for the function on the interval . 9. On what interval is the function 10. Find the interval on which increasing? is increasing. 11. Find the x-coordinate of the point of inflection of the function 12. How many points of inflection does the function 13. On what interval is the graph of have? concave downward? 14. Find the interval on which the graph of 15. Find the critical numbers: 3 2 (a) y = 2 . concave upward? 1 (b) y = 3 (7 ) (c) y = 92 16. Consider () = 3 12 + 2. (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. (d) Using the information in (a), (b), and (c) to sketch a graph of f. 17. Let f(x) = x5 + 3x2 5x 7, find the interval(s) where f is increasing and decreasing. 18. Find the value of the limit: 19. Find the value of the limit: 20. Find the value of the limit: 21. Find two numbers whose difference is 100 and whose product is a minimum. 22. The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares? 23. An open box is made from a 8 inch 8 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. What size squares should be cut out to create a box with maximum volume? 24. A farmer has 20 feet of fence, and he wishes to make from it a rectangular pen for his pig Wilbur, using a barn as one of the sides. In square feet, what is the maximum area possible for this pen? 25. If 1200 2 if material is available to make a box with a square base and open top, find the largest possible volume of the box. 26. A box with a square base and open top must have a volume of 32,000 3 . Find the dimensions of the box that minimize the amount of material used. 27. A company has a cost function units should it make to maximize its profit? and demand function . How many 28. Suppose that a baseball is tossed straight up and that its height as a function of time (in seconds) is given by the formula . What is the maximum height of the ball? 29. Find the point on the line y = 2x + 3 that is closest to the origin. 30. A cylindrical can is to be made to hold 1000 3 of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. 31. The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 2 , find the dimensions of the poster with the smallest area. 32. A rectangular storage container with an open top is to have a volume of 10 m : The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the dimensions of the container needed to minimize the cost. 33. Use Newton's method with the initial approximation root of the equation . to find , the second approximation to a Math 2211 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 4.1-4.8 Review SOLUTIONS 0, 8 No critical number 0, 3, 3 3 2 1 2 f(0) = 0 is a Local Maximum, f(3 ) = 9 is a Local Minimum, f( 3 ) = 9 is a Local Minimum 2 - 1 (1, 1) (, 1) 1 3 3 3 (2, ) (1, 1) 7 (a) C.N.: 0, 4 (b) C.N.: 0, (c) C.N.: None 4 (a) f is increasing on (, 2) (2, ) and decreasing on (2, 2). (b) f(2) = 18 is a Local Maximum, f(2) = 14 is a Local Minimum. (c) f is concave up on (0, ) and concave down on (-, 0). Inflection point (0, 2) 17. increasing on ( , 1.259) (.668, ); decreasing on (1.259, .668) 18. 0 1 19. 6 20. 2 21. 50 and 50 22. 128 4 4 23. 3 in x 3 in 24. 50 25. 4000 cm3 26. 40 x 40 x 20 27. 25 28. 256 feet 6 3 29. ( 5 , 5) 30. radius = 5.42 cm. and height = 10.84 cm. 31. 24 cm x 26 cm 32. 1.65 x 3.3 x 1.83 81 33. 40