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 .
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
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
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.
<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
51.
<ul><li>From the definition of a derivative, we have: </li></ul>DERIVS. OF TRIG. FUNCTIONS Equation 1
52.
<ul><li>Two of these four limits are easy to evaluate. </li></ul>DERIVS. OF TRIG. FUNCTIONS
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
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
55.
<ul><li>We can deduce the value of the remaining limit in Equation 1 as follows. </li></ul>DERIVS. OF TRIG. FUNCTIONS
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
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
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
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
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
63.
<ul><li>The Quotient Rule gives: </li></ul>Example 2 Solution: tan2 x + 1 = sec2 x
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
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:
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
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:
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
69.
<ul><ul><ul><li>THANK YOU </li></ul></ul></ul>
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