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Compilation of Problems in Calculus

1. What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu. m. if the error of the computed volume is not to exceed 0.02 cu. m?

2. Find the altitude of a cylinder of maximum volume which can be inscribed in a right circular cone of radius r and height h.

3. Find the approximate change in the volume of a cube of side “x” inches caused by increasing its side by 1%. 

4. Three sides of a trapezoid are each 8-cm long. How long is the fourth side when the area of the trapezoid has the greatest value? 

5. A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum.

6. A balloon is rising vertically over a point A on the ground at the rate of 15 ft/sec. A point B on the ground level with and 30 ft from A. When the balloon is 40 ft from A, at what rate is its distance from B changing? 

7. Find the point in the parabola y^2=4x at which the rate of change of the ordinate and abscissa are equal. 

8. Find the slope ofx^2+ y=8 at the point (2, 2).

9. Find the equation of the normal to x^2+y^2=1 at the point (2, 1). 

10. The depth of water in a cylindrical tank 4 m in diameter is increasing at the rate of 0.70 m/min. Find the rate at which the water is flowing into the tank.

11. Find the coordinates of the vertex of the parabola y=x^2-4x+1 if the slope of the tangent at the vertex is zero.

12. Find the minimum distance from the point (4, 2) to the parabola y^2=8x.

13. Two posts, one 8 m and the other 12 m high are 15 cm apart. If the posts are supported by a cable running from the top of the first post to a stake on the ground and then back to the top of the second post, find the distance from the lower post to the stake to use minimum amount of wire? 

14. What is the area of the largest rectangle that can be inscribed in a semi-circle of radius 10? 

15. Find the approximate increase in the volume of the sphere if the radius increases from 2 to 2.05 in one second.

16. Differentiate: ln(ln y) + ln y = ln x

17. The volume of the sphere is increasing at the rate of 6 cm^3/hr. At what rate is its surface area increasing when the radius is 50 cm? 

18. If the sum of two numbers is 4, find the minimum value of the sum of their cubes. 

19. Water is running out a conical funnel at the rate of 1 cu. inch per second. If the radius of the base of the funnel is 4 inches and the altitude is 8 inches, find the rate at which the water level is dropping when it is 2 inches from the top. 

20. Divide the number 120 into two parts such that the product of one and the square of the other is a maximum.

21. If y = 2x + sin 2x, find x if y’ = 0. 

22. What is the equation of the tangent to the curve y=x+5/x at the point (1, 3)?

23. If y=arc tan(ln(x)), find y’ at x = 1/e. 

24. Find the change in y = 2x – 3 if x changes from 3.3 to 3.5.

25. The radius of a sphere is r inches at time t seconds. Find the radius when the rates of increase of the surface area and the radius are numerically equal. 

26. Using the two existing corner sides of an existing wall, what is the maximum rectangular area that can be fenced by a fencing material 30 ft. long?

27. A man on a wharf 3.6 m above sea level is pulling a rope tied to a raft at 0.60 m/sec. How fast is the raft approaching the wharf when there are 6 m of rope out? 

28. The sum of four positive integers is 32. Find the greatest possible product of these four numbers.

29. Find the maximum point of y=x+1/x.

30. Find the height of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius 10 cm.

31. Find the area bounded by the curve defined by the equation x^2=8y and its latus rectum. 

32. Evaluate: The integral of (cos^8 3A dA) from 0 to pi/6.

33. Find the area bounded by the parabolas x^2-2y=0 and x^2+2y-8=0. 

34. Evaluate: The integral of (√(1-cos⁡x ) dx)

35. Find the area bounded by the curve y=9-x^2  and the x-axis. 

36. Evaluate: The integral of 3x/ex dx from 0 to 1.

37. Evaluate: The integral of tan2x dx

38. Find the value of (1 + i)^5. 

39. Simplify the expression: i^1997  + i^1999.

40. What is the quotient when 4 + 8i is divided by i^3? 

41. If (x+yi)(2-4i)=14-8i, find x. 

42. Find the angle between the planes 3x – y + z – 5 = 0 and x + 2y + 2z + 2 = 0 

43. Find the area of the geometric figure whose vertices are at (3, 0, 0), (3, 3, 0), (0, 0, 4) and (0, 3, 4). 

44. Find the length of the vector (2, 4, 4). 

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