There are around 40 problems in this review. If it possible to have solutions written out for each that would be awesome! Questions that contain similar solutions can be skipped. Therefore, a total yield of around 25-30 solutions should be written.

Review Problems: Chapter 4 Exam II(Sections 3.7-4.4) 1. An equilateral triangle is inscribed in a circle of radius . Express the circumference of the circle as a function of the x the length x of a side of the triangle. 2. A rectangle has one corner in quadrant I on the graph of = 16 2 , another at the origin, a third on the positive y-axis, and the fourth on the positive x-axis. a) Express the area A of the rectangle as a function of . b) For What value of is A largest? c) Express the perimeter of the rectangle as a function of . 3. Let (, ) be a point on the graph of = . a) Express the distance d form to the point (1,0) as a function of . b) For what value of is d smallest? 4. Find two positive numbers whose product is 100 and whose sum is a minimum. 5. A box with a square base and an open box must have a volume of 28,000 cm3. Find the dimensions of the box that minimize the amount of material used. 6. If 1200 cm2 of material is available to make a box with a square base and a open top, find the largest possible volume of the box. A rectangular container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $13 per square meter. Material for the sides costs $10 per square meter. Find the cost of the material for the cheapest such container. Round the result to nearest cent. 7. 8. 1 A rectangle in the first quadrant I has one corner on the line = 6 3 , another on the positive yaxis, another on the origin and the fourth on the positive x-axis. What are the dimension of the largest such rentangle? 9. Find the antiderivative of the function () = 54 3 +2 6 . 6 10. Find the antiderivative of the function () = 2 + 81.8 . 11. Find if () = 4 6 40 3 , (0) = 2, (0) = 1. 12. Find if () = 2 12, (0) = 9, (2) = 15. 13. Find the position of a particle given () = 10 sin + 3 cos , (0) = 0, (2) = 12 14. A stone is dropped off a cliff and hits the ground with a speed of 120ft/s. What is the height of the cliff? 10 15. Evaluate the integral | 5| by interpreting it in terms of areas. 0 16. The velocity function () = 2 2 8, 1 6 (in meter per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. 17. Express the limit lim =1[2( )2 + ( )] as a definite integral on the interval [2,7]. Evaluate. 1 Evaluate 1 0 3 64 1+ 20. 1 1 2 2 19. ( + ) 1 18. 45 21. 2 3 (2 + 1)5 23. sin 22. (2 + 5 ) 0 1 4 2 25. (12 cos(3) 4sin(2)) 24. ( + 1) 6 0 26. sin 1 2 5 27. | 3| 2 6 28. (1 + 2 4 ) 3 29. 6 2 3 31. 4 2 30. ( 2||) 1 32. 0 8 sec 2 tan 2 33. Find the derivative of () = (2 + 4 )5 . 1 34. Find the derivative of () = 1 2 3 . 35. Find the average value of the function () = 2 on [0,2]. Find c satisfying the conclusion of MVT for integrals. 36. Find the average value of the function () = (1+)2 on [1,6]. Find c satisfying the conclusion of MVT for 0 3 integrals. 37. Let () = 2 + 1, use = 5 rectangles to approximate the area on the interval [1,3]. Then integrate to find the exact area. 38. Find two positive numbers such that the sum of the first number squared and the second number is 54 and the product is a maximum. 39. A rectangle page is to contain 36 in2 of printed area. The margins on each side are 1 2 inches. Find the 1 dimensions of the page such that the least amount of paper is used. 2 Answers: 1. = 23 3 2. () = 16 3 , = 43 , 3 1 3.a) () = 2 + 1b) = 2 () = 32 + 2 2 2 4. 10,10 5. 38.3in x 38.3 in x 19.1 in 7. $ 250.90 8. 9 x 3 10. () = 2 + 20 7 2.8 + 13. () = 10 3 + 6 + 3 16. a) 10 , 3 b) 98 3 31. 12. () = 2 3 + 2 + 9 + 9 14. 225 ft 15. 25 1475 6 5 18. 9 256 5 21. 3 8(2+1)4 23. cos() + 1 24. 49 15 26. sec + 27. 2.5 29. 0 30. 3.5 32. 2 1 33. () = (2 + 4 )5 20. 156 7 1 2 1 2 + 2 + 2 + 5 11. () = 2 5 3 + 2 2 + + 2 7 25. 3 28. 9. () = 17. (2 2 + ) , 2 19. 22. 6. 4000cm3 1 2 + 3 6 + 33 3 34. () = 2 1 sin3 ( 2 ) 3 37. app. = 12.32, 1 35. = 3, = 32 3 38. 32, 36 3+21 6 + 3 36. = 14 , = 1 + 14 39. 9in by 9 in. 3