1. BUS-221 Quantitative Methods
LECTURE 4

2. Learning Outcome
Knowledge - Be familiar with basic mathematical techniques including: calculus (differential and integral)
Research - Retrieve and analyse information from directed sources for calculation and interpretation
 Mentation - Analyse business case studies and make decisions based on quantitative data.

3. Topics Calculus (integral and differential) Differentiation
Rate of Change
Fundamental Theorem of Calculus

4. Limits (1 of 9)

5. Limits (2 of 9)
Example 1 – Estimating a Limit from a Graph

6. Limits (3 of 9)
Example 1 – Continued

7. Continuity (1 of 5)
Example 1 – Applying the Definition of Continuity

8. The Derivative (1 of 6)
Below are examples of a tangent to a curve:
The slope of a curve at a point P is the slope, if it exists, of the tangent line at P.

9. The Derivative (2 of 6)
Example 1 – Finding the Slope of a Tangent Line

10. The Derivative (3 of 6)

11. The Derivative (4 of 6)
Example – Finding an Equation of a Tangent Line

12. The Derivative (5 of 6)
Example – A Function with a Vertical Tangent Line

13. The Derivative (6 of 6)
Example – Continuity and Differentiability

14. Rules for Differentiation (1 of 7)
Below are some rules for differentiation: BASIC RULE 1 Derivative of a Constant:
BASIC RULE 2 Derivative of xn:
COMBINING RULE 1 Constant Factor Rule:
COMBINING RULE 2 Sum or Difference Rule:

15. Rules for Differentiation (2 of 7)
Example 1 – Derivatives of Constant Functions

16. Rules for Differentiation (3 of 7)
Example – Rewriting Functions in the Form xa

17. Rules for Differentiation (4 of 7)
Example – Differentiating Sums and Differences of Functions

18. Rules for Differentiation (5 of 7)
Example – Continued

19. Rules for Differentiation (6 of 7)
Example – Continued

20. Rules for Differentiation (7 of 7)
Example – Finding an Equation of a Tangent Line

21. The Derivative as a Rate of Change (1 of 7)
Example 1 – Finding Average Velocity and Velocity

22. The Derivative as a Rate of Change (2 of 7)
Example 1 – Continued

23. The Derivative as a Rate of Change (3 of 7)
Example – Finding a Rate of Change

24. The Derivative as a Rate of Change (4 of 7)
Example – Rate of Change of Volume A spherical balloon is being filled with air. Find the rate of change of the volume of air in the balloon with respect to its radius. Evaluate this rate of change when the radius is 2 ft.

25. The Derivative as a Rate of Change (5 of 7)
Applications of Rate of Change to Economics

26. The Derivative as a Rate of Change (6 of 7)
Example – Marginal Cost

27. The Derivative as a Rate of Change (7 of 7)
Example – Relative and Percentage Rates of Change

28. Elasticity of Demand (1 of 2)

29. Elasticity of Demand (2 of 2)
Example – Finding Point Elasticity of Demand

30. Implicit Differentiation (1 of 3)
Implicit Differentiation Procedure

31. Implicit Differentiation (2 of 3)
Example 1 – Implicit Differentiation

32. Implicit Differentiation (3 of 3)
Example – Implicit Differentiation

33. Concavity (1 of 6)

34. Concavity (2 of 6)
Rule 1 Criteria for Concavity

35. Concavity (3 of 6)
Example – Testing for Concavity

36. Concavity (4 of 6)
Example 1 – Continued

37. Concavity (5 of 6)
Example – A Change in Concavity with No Inflection Point

38. Concavity (6 of 6)
Example – Continued

39. The Second-Derivative Test (1 of 3)

40. The Second-Derivative Test (2 of 3)
Example 1 – Second-Derivative Test

41. The Second-Derivative Test (3 of 3)
Example 1 – Continued

42. Differentials (1 of 4)
Example 1 – Computing a Differential

43. Differentials (2 of 4)
Example – Using the Differential to Estimate a Change in a Quantity A governmental health agency examined the records of a group of individuals who were hospitalized with a particular illness. It was found that the total proportion P that are discharged at the end of t days of hospitalization is given by
Use differentials to approximate the change in the proportion discharged if t changes from 300 to 305.

44. Differentials (3 of 4)
Example – Continued

45. The Infinite Integral (1 of 7)

46. The Infinite Integral (2 of 7)
Example 1 – Finding an Indefinite Integral
Table 14.1 Elementary Integration Formulas

47. The Infinite Integral (5 of 7)
Example – Indefinite Integral of a Sum and Difference

48. The Infinite Integral (6 of 7)
Example – Using Algebraic Manipulation to Find an Indefinite Integral

49. The Infinite Integral (7 of 7)
Example – Continued

50. Integration with Initial Conditions (2 of 5)
Example – Income and Education

51. Integration with Initial Conditions (3 of 5)
Example – Continued

52. Integration with Initial Conditions (4 of 5)
Example – Finding Cost from Marginal Cost

53. Integration with Initial Conditions (5 of 5)
Example – Continued

54. Techniques of Integration (1 of 2)
Example 1 – Preliminary Division before Integration

55. The Definite Integral (1 of 6)

56. The Definite Integral (2 of 6)

57. The Fundamental Theorem of Calculus (1 of 5)

58. The Fundamental Theorem of Calculus (2 of 5)
Properties of the Definite Integral

59. The Fundamental Theorem of Calculus (3 of 5)
Example 1 – Applying the Fundamental Theorem

60. The Fundamental Theorem of Calculus (4 of 5)
Example – Evaluating Definite Integrals

61. The Fundamental Theorem of Calculus (5 of 5)
Example 5 – Finding a Change in Function Values by Definite Integration