1. Midterm Review Calculus

2. UNIT 0

3. Page  3 Determine whether is rational or irrational. Determine whether the given value of x satisfies the inequality: a.) x = -2b.) x = 0c.) x = d.) x = -6 RATIONAL SATISFIESDOES NOT SATISFY SATISFIES

4. Page  4 1.)2.) 3.)4.) Solve each inequality: x ≥ 3 -1 < x < 7

5. Page  5 Given the interval [-3, 7], find: a.) the distance between -3 and 7 b.) the midpoint of the interval c.) Use absolute value to describe this interval d = 10 Midpoint = 2

6. Page  6 Simplify each:

7. Page  7 Remove all possible factors from the radical: Complete the factorization:

8. Page  8 1.)2.) 3.) Factor each completely:

9. Page  9 Use the rational zero theorem to find all real roots of: Possible Rational Zeros: ±1, ±2, ±3, ±6 So -1, 2, and 3 are all roots

10. Page  10 Combine terms and simplify each:

11. Page  11 Combine terms and simplify each:

12. Page  12 Rationalize the denominator:

13. UNIT 1

14. Page  14 Find the distance between (3, 7) and (4, -2) Find the midpoint of the line segment joining (0, 5) and (2, 1) Determine whether the points (0, -3), (2, 5), and (-3, -15) are collinear. Midpoint (1, 3) All points are collinear

15. Page  15 Find x so that the distance between (0, 3) and (x, 5) is 7

16. Page  16 Sketch the graph of each:

17. Page  17 Write the equation of the circle in standard form and sketch it: Find the points of intersection of the graphs of: (0, -5) and (4, -3)

18. Page  18 Find the general equation of the line given certain information: a.) (7, 4) and (6, -2) b.) (-2, -1) and slope = ⅔

19. Page  19 Find the general equation of the line given certain information: a.) (6, -8) and undefined slope b.) (0, 3) and perpendicular to 2x – 5y = 7

20. Page  20 f(3)f(-6) f(x – 5)f(x + Δx) Given find the following:

21. Page  21 Find the domain and range of: Given and find: Domain: (-∞, 3] Range: [0, ∞)

22. Page  22 Given find

23. Page  23 1.)2.) 3.)4.) Find each limit:

24. Page  24 Find the

25. Page  25 Find the discontinuities of each and tell which are removable. x = ±8 x = 8 is removable x = 3 is a non-removable discontinuity

26. Page  26 Sketch the graph: Hole @ x = 2

27. UNIT 2

28. Page  28 Find the derivative of each: 1.) 2.)

29. Page  29 Use the derivative to find the equation of the tangent line to the graph of f(x) at the point (6, 2)

30. Page  30 Find f’(x) for each f(x) 1.) 2.) 3.)

31. Page  31 Find the average rate of change of f(x) over the interval [0. 2]. Compare this to the instantaneous rate of change at the endpoints of the interval. Average rate of change: 4 Instantaneous rates of change:

32. Page  32 Given the cost function C(x), find the marginal cost of producing x units. Marginal cost: 4.31 – 0.0002x

33. Page  33 Find f’(x) for each f(x) 1.) 2.) 3.)

34. Page  34 Find f’(x) for each f(x) 1.) 2.) 3.)

35. Page  35 Find the derivative of each: 1.) 2.)

36. Page  36 1.) Given f(x), find f’’’(x) 2.) Given f(x), find f’’’’(x)

37. Page  37 Use implicit differentiation to find 1.) 2.)

38. Page  38 Use implicit differentiation to find 1.) 2.)

39. Page  39 Let y = 3x 2. Find when x = 2 and = 5

40. Page  40 The area A of a circle is increasing at a rate of 10 in. 2 /min. Find the rate of change of the radius r when r = 4 inches.

41. Page  41 The volume of a cone is. Find the rate of change of the height when :

42. UNIT 3.1-3.4

43. Page  43 Find the critical numbers and the intervals on which f(x) is increasing or decreasing for f(x): Increasing: (-∞, 0) U (4, ∞) Decreasing: (0, 4)

44. Page  44 Find the critical numbers and the intervals on which f(x) is increasing or decreasing for f(x): Increasing: (-∞, ⅔) Decreasing: (⅔, 1)

45. Page  45 Find the relative extrema of f(x) Relative Minimum: (2, -45)

46. Page  46 Find the relative extrema of f(x) Relative Minimum: (-3, 0)

47. Page  47 Find the absolute extrema of f(x) on [0, 5] Abs. Max: (5, 0) Abs. Min: (2, -9)

48. Page  48 Find the points of inflection of f(x) No Inflections Points

49. Page  49 Find the points of inflection of f(x) Points of Inflection:

50. Page  50 Find two positive numbers who product is 200 such that the sum of the first plus three times the second is a minimum. First number: Second number:

51. Page  51 Three rectangular fields are to be enclosed by 3000 feet of fencing, as shown below. What dimensions should be used so that the enclosed area will be a maximum? y x x x 3x = 750 feet, y = 375 feet