1. 5.1 Accumulating Change: Introduction to results of change
Business Calculus II
5.1
Accumulating Change: Introduction to results of change

2. Accumulated Change
If the rate-of-change function f’ of a quantity is continuous over an interval a<x<b, the accumulated change in the quantity between input values of a and b is the area of the region between the graph and horizontal axis, provided the graph does not crosses the horizontal axis between a and b.
If the rate of change is negative, then the accumulated change will be negative.
Example:
Positive- distance travel
Negative-water draining from the pool

3. 5.1 – Accumulated Distance (PAGE 319)

4. Accumulated Change involving Increase and decrease
Calculate positive region (A)
Calculate negative region (B)
Then combine the two for overall change

5. Rate of Change (ROC) Function Behavior
Maximum
Rate of Change (ROC) Function Behavior
Minimum
Positive Slope
Negative Slope
Positive Slope
Zero
Zero

6. Rate of Change (ROC) Function Behavior
Inflection Point
Concave Down
Decreasing
Concave Up
Increasing

7. Problems 2, 6, 7, 12 (pages )

8. 5.2 Limits of Sums and the Definite Integral
Business Calculus II
5.2
Limits of Sums and the Definite Integral

9. Approximating Accumulated Change
Not always graphs are linear!
Left Rectangle approximation
Right Rectangle approximation
Midpoint Rectangle approximation

10. Left Rectangle approximation

11. Sigma Notation
When xm, xm+1, …, xn are input values for a function f and m and n are integers when m<n, the sum f(xm)+f(xm+1)+….f(xn) can be written using the greek capital letter sigma () as

12. Right Rectangle approximation

13. Mid-Point Rectangle approximation

14. Area Beneath a Curve Area as a Limit of Sums
Let f be a continuous nonnegative function from a to b. The area of the region R between the graph of f and x-axis from a to b is given by the limit
Where xi is the midpoint of the ith subinterval of length x= (b-a)/n between a and b.

15. Page 334- Quick Example
Calculator Notation for midpoint approximation: Sum(seq(function * x, x, Start, End, Increment)
Start: a + ½ x
End: b - ½ x
Increment: x

16. Left rectangle
Calculator Notation : Sum(seq(function * x, x, Start, End, Increment)
Start: a
End: b - x
Increment: x

17. Right Rectangle
Calculator Notation: Sum(seq(function * x, x, Start, End, Increment)
Start: a + x
End: b
Increment: x

18. Related Accumulated Change to signed area
Net Change in Quantity
Calculate each region and then combine the area.

19. Definite Integral
Let f be a continuous function defined on interval from a to b. the accumulated change (or definite Integral) of f from a to b is Where xi is the midpoint of the ith subinterval of length x= (b-a)/n between a and b.

20. Problems 2, 8 (pages )

21. 5.3 Accumulation Functions
Business Calculus II
5.3
Accumulation Functions

22. Accumulation Function
The accumulation function of a function f, denoted by gives the accumulation of the signed area between the horizontal axis and the graph of f from a to x. The constant a is the input value at which the accumulation is zero, the constant a is called the initial input value.

23. 2. Velocity (page 350) x 1 2 3 4 5 6 7 8 9 10
Area
Acc. Area

24. 4. Rainfall (page 351)

25. Using Concavity to refine the sketch of an accumulation Function (Page 348)
Faster
Slower
Increase
decrease
decrease
Increase
Slower
Faster

26. Graphing Accumulation Function using F’
When F’ Graph has x-intercept, then you have Max/Min/inflection point in accumulation graph
How to identify the critical value(s):
MAX in Accumulation graph:
When F’ graph changes from Positive to negative
MIN in Accumulation graph:
When f’ graph changes from negative to positive
Inflection point in accumulation graph:
When F’ touches the x-axis Or
You have MAX/MIN in F’ graph

27. Graphing Accumulation Function using F’
Max: Positive to negative
Positive F’
x-intercept, MAX – in Accumulation graph
Negative F’

28. Graphing Accumulation Function using F’
Min: negative to Positive
Positive F’
x-intercept, MIN – in Accumulation graph
  Negative F’

29. Graphing Accumulation Function using F’
Inflection Point: F’ Touches the x-axis
x-intercept, MIN – in Accumulation graph

30. Graphing Accumulation Function using F’
Inflection Point: inflection point in F’, also appears as inflection point in accumulation graph
Inflection Points in F’

31. WHAT WE HAVE COMBINE
INF
INF
MAX
MIN
INF
INF
INF

32. Positive area
Start at zero

33. 10-Sketch

34. 12-sketch

35. 14-sketch

36. Business Calculus II
5.4
Fundamental Theorem

37. Fundamental Theorem of Calculus (Part I)
For any continuous function f with input x, the derivative of in term use of x: FTC Part 2 appears in Section 5.6.

38. Anti-derivative Reversal of the derivative process
Let f be a function of x . A function F is called an anti-derivative of f if
That is, F is an anti-derivative of f if the derivative of F is f.

39. General and Specific Anti-derivative
For f, a function of x and C, an arbitrary constant,
is a general anti-derivative of f
When the constant C is known, F(x) + C is a specific anti-derivative.

40. Simple Power Rule for Anti-Derivative

41. More Examples:

42. Constant Multiplier Rule for Anti-Derivative

43. Sum Rule and Difference Rule for Anti-Derivative

44. Example:

45. Connection between Derivative and Integrals
For a continuous differentiable function fwith input variable x,

46. Example:

47. Problem: 2,12,14,16,20,22,24,37

48. 5.5 Anti-derivative formulas for Exponential, LN
Business Calculus II
5.5 Anti-derivative formulas for
Exponential, LN

49. 1/x(or x-1) Rule for Anti-derivative
ex Rule for Anti-derivative
ekx Rule for Anti-derivative

50. Exponential Rule for Anti-derivative
Natural Log Rule for Anti-derivative
Please note we are skipping Sine and Cosine Models

51. Example

52. Example (16 – page 373):

53. Problems: 2, 6, 8, 10, 20, 24 (page )

54. 5.6 The definite Integral - Algebraically
Business Calculus II
5.6 The definite Integral - Algebraically

55. The fundamental theorem of Calculus (Part 2) – Calculating the Definite Integral (Page 375)
If f is continuous function from a to b and F is any anti-derivative of f, then
Is the definite integral of f from a to b.
Alternative notation

56. Sum Property of Integrals
Where b is a number between a and c

57. Definite Integrals as Areas
For a function f that is non-negative from a to b
= the area of the region between f and the x-axis from a to b

58. Definite Integrals as Areas
For a function f that is negative from a to b
= the negative of the area of the region between f and the x-axis from a to b

59. Definite Integrals as Areas
For a general function f defined over an interval from a to b
= the sum of the signed area of the region between f and the x-axis from a to b
= ( the sum of the areas of the region above the a-axis) minus (the sum of the area of the region below the x-axis)

60. Problems: 10, 14, 18, 20, 22

61. 5.7 Difference of accumulation change
Business Calculus II
5.7 Difference of accumulation change

62. Area of the region between two curves
If the graph of f lies above the graph of g from a to b, then the area of the region between the two curves is given by

63. Difference between accumulated Changes
If f and g are two continuous rates of change functions, the difference between the accumulated change of f from a to b and the accumulated change of g between a and b is the accumulated change in the difference between f-g

64. Problems: 2, 6, 10, 12, 14

65. 5.8 Average Value and Average rate of change
Business Calculus II
5.8 Average Value and Average rate of change

66. Average Value
If f is continuous function from a to b, the average value of f from a to b is

67. The average value of the rate of change
If f’ is a continues rate of change function from a to b, the average value of f’ from a to b is given as
Where f is a anti-derivative of f’.

68. Problems: 2, 6, 10, 18