Limits and Derivatives

1. Limits and Derivatives

2. Concept of a Function

3. y = x2 y is a function of x, and the relation y = x2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y.

4. Since the value of y depends on a given value of x, we call y the dependent variable and x the independent variable and of the function y = x2.

8. Notation for a Function : f(x)

19. The Idea of Limits

20. The Idea of Limits Consider the function

21. The Idea of Limits Consider the function

22. y 2 x O The Idea of Limits Consider the function

23. If a function f(x) is a continuous at x0, then . approaches to, but not equal to

24. The Idea of Limits Consider the function

25. The Idea of Limits Consider the function

26. does not exist.

27. A function f(x) has limit l at x0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x0. We write

28. Theorems On Limits

29. Theorems On Limits

30. Theorems On Limits

31. Theorems On Limits

32. Exercise 12.1 P.7

33. Limits at Infinity

34. Limits at Infinity Consider

35. Generalized, if then

36. Theorems of Limits at Infinity

37. Theorems of Limits at Infinity

38. Theorems of Limits at Infinity

39. Theorems of Limits at Infinity

40. Exercise 12.2 P.13

41. Theorem where θ is measured in radians. All angles in calculus are measured in radians.

42. Exercise 12.3 P.16

43. The Slope of the Tangent to a Curve

44. The Slope of the Tangent to a Curve The slope of the tangent to a curve y = f(x) with respect to x is defined as provided that the limit exists.

45. Exercise 12.4 P.18

46. Increments The increment △x of a variable is the change in x from a fixed value x = x0 to another value x = x1.

47. For any function y = f(x), if the variable x is given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence thee is a corresponding increment of y(△y) such that △y = f(x0 + △x) – f(x0).

48. Derivatives The derivative of a function y = f(x) with respect to x is defined as provided that the limit exists. (A) Definition of Derivative.

49. The derivative of a function y = f(x) with respect to x is usually denoted by

50. The process of finding the derivative of a function is called differentiation. A function y = f(x) is said to be differentiable with respect to x at x = x0 if the derivative of the function with respect to xexists at x = x0.