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# Limits And Derivative

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### Limits And Derivative

1. 1. <ul><li>Done BY, </li></ul><ul><li>Achuthan </li></ul><ul><li>xi b </li></ul><ul><li>k.v.pattom </li></ul>
2. 2. Limits and Derivatives
3. 3. Concept of a Function
4. 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. 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 .
6. 9. Notation for a Function : f ( x )
7. 20. The Idea of Limits
8. 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 )
9. 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
10. 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
11. 24. If a function f ( x ) is a continuous at x 0 , then . approaches to, but not equal to
12. 25. Consider the function The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 g ( x )
13. 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
14. 27. does not exist.
15. 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
16. 29. Theorems On Limits
17. 30. Theorems On Limits
18. 31. Theorems On Limits
19. 32. Theorems On Limits
20. 33. Limits at Infinity
21. 34. Limits at Infinity Consider
22. 35. Generalized, if then
23. 36. Theorems of Limits at Infinity
24. 37. Theorems of Limits at Infinity
25. 38. Theorems of Limits at Infinity
26. 39. Theorems of Limits at Infinity
27. 40. Theorem where θ is measured in radians . All angles in calculus are measured in radians.
28. 41. The Slope of the Tangent to a Curve
29. 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.
30. 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 .
31. 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 ) .
32. 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.
33. 46. The derivative of a function y = f ( x ) with respect to x is usually denoted by
34. 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 .
35. 48. The value of the derivative of y = f ( x ) with respect to x at x = x 0 is denoted by or .
36. 49. To obtain the derivative of a function by its definition is called differentiation of the function from first principles .
37. 50. <ul><li>Let’s sketch the graph of the function f ( x ) = sin x , it looks as if the graph of f’ may be the same as the cosine curve. </li></ul>DERIVATIVES OF TRIGONOMETRIC FUNCTIONS Figure 3.4.1, p. 149
38. 51. <ul><li>From the definition of a derivative, we have: </li></ul>DERIVS. OF TRIG. FUNCTIONS Equation 1
39. 52. <ul><li>Two of these four limits are easy to evaluate. </li></ul>DERIVS. OF TRIG. FUNCTIONS
40. 53. <ul><li>Since we regard x as a constant when computing a limit as h -> 0, we have: </li></ul>DERIVS. OF TRIG. FUNCTIONS
41. 54. <ul><li>The limit of (sin h )/ h is not so obvious. </li></ul><ul><li>In Example 3 in Section 2.2, we made the guess—on the basis of numerical and graphical evidence—that: </li></ul>DERIVS. OF TRIG. FUNCTIONS Equation 2
42. 55. <ul><li>We can deduce the value of the remaining limit in Equation 1 as follows. </li></ul>DERIVS. OF TRIG. FUNCTIONS
43. 56. DERIVS. OF TRIG. FUNCTIONS Equation 3
44. 57. <ul><li>If we put the limits (2) and (3) in (1), we get: </li></ul><ul><li>So, we have proved the formula for sine, </li></ul>DERIVS. OF TRIG. FUNCTIONS Formula 4
45. 58. <ul><li>Differentiate y = x 2 sin x . </li></ul><ul><ul><li>Using the Product Rule and Formula 4 , we have: </li></ul></ul>Example 1 DERIVS. OF TRIG. FUNCTIONS Figure 3.4.3, p. 151
46. 59. <ul><li>Using the same methods as in the proof of Formula 4, we can prove: </li></ul>Formula 5 DERIV. OF COSINE FUNCTION
47. 60. DERIV. OF TANGENT FUNCTION Formula 6
48. 61. <ul><li>We have collected all the differentiation formulas for trigonometric functions here. </li></ul><ul><ul><li>Remember, they are valid only when x is measured in radians. </li></ul></ul>DERIVS. OF TRIG. FUNCTIONS
49. 62. <ul><li>Differentiate </li></ul><ul><li>For what values of x does the graph of f have a horizontal tangent? </li></ul>Example 2 DERIVS. OF TRIG. FUNCTIONS
50. 63. <ul><li>The Quotient Rule gives: </li></ul>Example 2 Solution: tan2 x + 1 = sec2 x
51. 64. <ul><li>Find the 27th derivative of cos x . </li></ul><ul><ul><li>The first few derivatives of f ( x ) = cos x are as follows: </li></ul></ul>Example 4 DERIVS. OF TRIG. FUNCTIONS
52. 65. <ul><ul><li>We see that the successive derivatives occur in a cycle of length 4 and, in particular, f ( n ) ( x ) = cos x whenever n is a multiple of 4. </li></ul></ul><ul><ul><li>Therefore, f (24) ( x ) = cos x </li></ul></ul><ul><ul><li>Differentiating three more times, we have: f (27) ( x ) = sin x </li></ul></ul>Example 4 Solution:
53. 66. <ul><li>Find </li></ul><ul><ul><li>In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7: </li></ul></ul>Example 5 DERIVS. OF TRIG. FUNCTIONS
54. 67. <ul><li>If we let θ = 7 x , then θ -> 0 as x -> 0. So, by Equation 2, we have: </li></ul>Example 5 Solution:
55. 68. <ul><li>Calculate . </li></ul><ul><ul><li>We divide the numerator and denominator by x : by the continuity of cosine and Eqn. 2 </li></ul></ul>Example 6 DERIVS. OF TRIG. FUNCTIONS
56. 69. <ul><ul><ul><li>THANK YOU </li></ul></ul></ul>