2. Limits and Derivatives

3. Concept of a Function

4. y is a function of x, and the relation y = x 2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y. y = x 2

5. 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 = x 2.

9. Notation for a Function : f(x)

20. The Idea of Limits

21. Consider the function The Idea of Limits x1.91.991.9991.999922.00012.0012.012.1 f(x)f(x)

22. Consider the function The Idea of Limits x1.91.991.9991.999922.00012.0012.012.1 f(x)f(x)3.93.993.9993.9999un- defined 4.00014.0014.014.1

23. Consider the function The Idea of Limits x1.91.991.9991.999922.00012.0012.012.1 g(x)g(x)3.93.993.9993.999944.00014.0014.014.1 x y O 2

24. If a function f(x) is a continuous at x 0, then. approaches to, but not equal to

25. Consider the function The Idea of Limits x-4-3-201234 g(x)g(x)

26. Consider the function The Idea of Limits x-4-3-201234 h(x)h(x) un- defined 1234

27. does not exist.

28. A function f(x) has limit l at x 0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x 0. We write

29. Theorems On Limits

33. Exercise 12.1 P.7

34. Limits at Infinity

35. Consider

36. Generalized, if then

37. Theorems of Limits at Infinity

41. Exercise 12.2 P.13

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

43. Exercise 12.3 P.16

44. The Slope of the Tangent to a Curve

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

46. Exercise 12.4 P.18

47. Increments The increment △ x of a variable is the change in x from a fixed value x = x 0 to another value x = x 1.

48. For any function y = f(x), if the variable x is given an increment △ x from x = x 0, then the value of y would change to f(x 0 + △ x) accordingly. Hence thee is a corresponding increment of y( △ y) such that △ y = f(x 0 + △ x) – f(x 0 ).

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

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

51. 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 = x 0 if the derivative of the function with respect to x exists at x = x 0.

52. The value of the derivative of y = f(x) with respect to x at x = x 0 is denoted by or.

53. To obtain the derivative of a function by its definition is called differentiation of the function from first principles.

54. Exercise 12.5 P.21