Limits And Derivative
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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.
Limits And Derivative
- 1. Done BY, Achuthan xi b k.v.pattom
- 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
- 34. Limits at Infinity Consider
- 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 .
- 50. 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. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS Figure 3.4.1, p. 149
- 51. From the definition of a derivative, we have: DERIVS. OF TRIG. FUNCTIONS Equation 1
- 52. Two of these four limits are easy to evaluate. DERIVS. OF TRIG. FUNCTIONS
- 53. Since we regard x as a constant when computing a limit as h -> 0, we have: DERIVS. OF TRIG. FUNCTIONS
- 54. The limit of (sin h )/ h is not so obvious. In Example 3 in Section 2.2, we made the guess—on the basis of numerical and graphical evidence—that: DERIVS. OF TRIG. FUNCTIONS Equation 2
- 55. We can deduce the value of the remaining limit in Equation 1 as follows. DERIVS. OF TRIG. FUNCTIONS
- 56. DERIVS. OF TRIG. FUNCTIONS Equation 3
- 57. If we put the limits (2) and (3) in (1), we get: So, we have proved the formula for sine, DERIVS. OF TRIG. FUNCTIONS Formula 4
- 58. Differentiate y = x 2 sin x . Using the Product Rule and Formula 4 , we have: Example 1 DERIVS. OF TRIG. FUNCTIONS Figure 3.4.3, p. 151
- 59. Using the same methods as in the proof of Formula 4, we can prove: Formula 5 DERIV. OF COSINE FUNCTION
- 60. DERIV. OF TANGENT FUNCTION Formula 6
- 61. We have collected all the differentiation formulas for trigonometric functions here. Remember, they are valid only when x is measured in radians. DERIVS. OF TRIG. FUNCTIONS
- 62. Differentiate For what values of x does the graph of f have a horizontal tangent? Example 2 DERIVS. OF TRIG. FUNCTIONS
- 63. The Quotient Rule gives: Example 2 Solution: tan2 x + 1 = sec2 x
- 64. Find the 27th derivative of cos x . The first few derivatives of f ( x ) = cos x are as follows: Example 4 DERIVS. OF TRIG. FUNCTIONS
- 65. 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. Therefore, f (24) ( x ) = cos x Differentiating three more times, we have: f (27) ( x ) = sin x Example 4 Solution:
- 66. Find In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7: Example 5 DERIVS. OF TRIG. FUNCTIONS
- 67. If we let θ = 7 x , then θ -> 0 as x -> 0. So, by Equation 2, we have: Example 5 Solution:
- 68. Calculate . We divide the numerator and denominator by x : by the continuity of cosine and Eqn. 2 Example 6 DERIVS. OF TRIG. FUNCTIONS
- 69. THANK YOU