The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.

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 x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f ( x )

22. Consider the function The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f ( x ) 3.9 3.99 3.999 3.9999 un-defined 4.0001 4.001 4.01 4.1

23. Consider the function The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 g ( x ) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1 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 -2 -1 0 1 2 3 4 g ( x )

26. Consider the function The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 h ( x ) -1 -1 -1 -1 un-defined 1 2 3 4

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

30. Theorems On Limits

31. Theorems On Limits

32. Theorems On Limits

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

41. The Slope of the Tangent to a Curve

42. 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.

43. 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 .

44. 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 ) .

45. 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.

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

47. 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 .

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

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

56. DERIVS. OF TRIG. FUNCTIONS Equation 3

60. DERIV. OF TANGENT FUNCTION Formula 6