Limits And Derivative slayerix

Done BY, Achuthan xi b k.v.pattom

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

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 .

Notation for a Function : f ( x )

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 )

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

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

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

Consider the function The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 g ( x )

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

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

Theorems of Limits at Infinity

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

The Slope of the Tangent to a Curve

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.

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 .

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 ) .

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.

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

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 .

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

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

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

From the definition of a derivative, we have: DERIVS. OF TRIG. FUNCTIONS Equation 1

Two of these four limits are easy to evaluate. DERIVS. OF TRIG. FUNCTIONS

Since we regard x as a constant when computing a limit as h -> 0, we have: DERIVS. OF TRIG. FUNCTIONS

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

We can deduce the value of the remaining limit in Equation 1 as follows. DERIVS. OF TRIG. FUNCTIONS

DERIVS. OF TRIG. FUNCTIONS Equation 3

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

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

Using the same methods as in the proof of Formula 4, we can prove: Formula 5 DERIV. OF COSINE FUNCTION

DERIV. OF TANGENT FUNCTION Formula 6

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

Differentiate For what values of x does the graph of f have a horizontal tangent? Example 2 DERIVS. OF TRIG. FUNCTIONS

The Quotient Rule gives: Example 2 Solution: tan2 x + 1 = sec2 x

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

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:

Find In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7: Example 5 DERIVS. OF TRIG. FUNCTIONS

If we let θ = 7 x , then θ -> 0 as x -> 0. So, by Equation 2, we have: Example 5 Solution:

Calculate . We divide the numerator and denominator by x : by the continuity of cosine and Eqn. 2 Example 6 DERIVS. OF TRIG. FUNCTIONS

Limits And Derivative slayerix

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