Calculus

1. Calculus • Limits • Continuity of Functions • Differentiation • Application of differentiation • Integration

2. Limits • Example from tangent line • Example from Decimal expansion of 1/3=0.33333…. • Definition of limits – get closer but do not touch • Few more examples of limits

3. Limit of series

4. Limit of series

5. Limit of series

6. Limit of series

7. Limit of series

8. Limit of series

9. Limit of series

10. Limit of series

11. Limits • Example from geometric series • Limit of a series • Definition of limits – get closer but do not touch • Few more examples of limits

12. Some Important Limits • sin(x)/x at x=0 • Definition of e • Limit of 1/x at x=0 • Limit of 1/(x*x) at x=0 • more examples

13. A practical Example • Interest compounded continuously • S = P[(1 + r/k)^(kn)] where n– no of years, k is no of times interest is compounded in a year, other variables have usual meaning. • Let k -> infinity. • S = P* e^(rn) • E.g. a trust fund is set by a single payment so that at the end of 20 years, there is $25000 in the fund. If the interest is calculated at a rate of 7% compounded continuously, find the amount of money to be invested.

14. Continuity • Definition • f(x)=1/x if x‚0; is f(x) continuous at x=0? • f(x)=1/x if x‚0 otherwise f(x)=0; is f continuous at x=0? • f(x)=x^2/x if x‚0 otherwise f(x)=1; is f continuous at x=0? • f(x)=x^2/x if x‚0 otherwise f(x)=0; is f continuous at x=0?

15. Feeling Continuity

16. Feeling Continuity

17. Feeling Continuity

18. Feeling Continuity

19. Feeling Continuity

20. Feeling Continuity

21. Feeling Continuity

22. Feeling Continuity

23. Feeling Continuity

24. Feeling Continuity

25. A discontinuous Function

26. A discontinuous Function

27. Differentiation • Algebraic motivation – rate of change

28. Algebraic Motivation

29. Algebraic Motivation

30. Algebraic Motivation

31. Algebraic Motivation

32. Algebraic Motivation

33. Differentiation • Algebraic motivation – rate of change • Geometric motivation – limiting position of secant

34. Geometric Motivation

35. Geometric Motivation

36. Geometric Motivation

37. Geometric Motivation

38. Geometric Motivation

39. Differentiation • Definition • Rules of Differentiation • Examples • Differentiable Functions

40. Feeling Differentiation

41. Feeling Differentiation

42. Feeling Differentiation

43. Some more Differentiation • Derivatives of Logarithmic functions • Derivatives of Exponential functions • Derivative of implicit functions • examples • Higher Order Derivatives

44. Multivariate Calculus • Functions of Several Variables • E.g. The marketing quantity may depend on the own price and price of competition. • Represented in a 3-D plane rather than a line. • Important in complex dependency models

45. Partial Differentiation • Rate of change of function when one variable changes • example

46. Applications of Partial Derivatives • E.g. A company manufactures two types of products A and B. Suppose the joint cost function for producing x pairs of A and y pairs of B is • C = 0.06x*x + 65x + 75y + 1000 • Determine the marginal costs wrt x and y and evaluate these when x = 100 and y = 50. Interpret.

47. Applications of Partial Derivatives • E.g. MBA compensation : In a study of success among MBAs, it was calculated that current annual compensation was given by • Z = 10,990 + 1120x + 873y, where x and y are number of years of work ex prior and post MBA. Find the partial derivative wrt x and y and interpret.

48. Higher Order Partial Derivatives • When a function depends on two variables and we desire to find the sensitivity wrt both at the same time • Derivative can be taken wrt same variable more than once also to determine the nature of the rate of change • example

49. Application of Differentiation • Extremal Problems – maxima and minima • Curve Sketching • Mathematical Modelling

50. Maxima and Minima