Limits And Derivative

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Limits and Derivatives

Concept of a Function

y is a function of x, and the relation y = x2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y.

y = x2

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 = x2.

Notation for a Function : f(x)

The Idea of Limits

Consider the function

The Idea of Limits

2

4)(

2

x

xxf

x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1

f(x)


The Idea of Limits

2

4)(

2

x

xxf

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


The Idea of Limits 2)( xxg

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

2)( xxg

x

y

O

2

If a function f(x) is a continuous at x0,

then . )()(lim 00

xfxfxx

4)(lim2

xfx

4)(lim2

xgx

approaches to, but not equal to


The Idea of Limits

x

xxh )(

x -4 -3 -2 -1 0 1 2 3 4

g(x)


The Idea of Limits

x

xxh )(

x -4 -3 -2 -1 0 1 2 3 4

h(x) -1 -1 -1 -1 un-defined

1 2 3 4

1)(lim0

xhx

1)(lim0

xhx

)(lim0

xhx does not

exist.

A function f(x) has limit l at x0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x0. We write

lxfxx

)(lim0

Theorems On Limits

Limits at Infinity

Limits at Infinity

Consider1

1)(

2

xxf

Generalized, if

)(lim xfx

then

0)(

lim xf

kx

Theorems of Limits at Infinity

Theorem

where θ is measured in radians.

All angles in calculus are measured in radians.

1sin

lim0

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.

x

xfxxf

x

yAT

xx

)()(limlim of Slope

00

Increments

The increment △x of a variable is the change in x from a fixed value x = x0 to another value x = x1.

For any function y = f(x), if the variable x is given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence thee is a corresponding increment of y(△y) such that △y = f(x0 + △x) –

f(x0).

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.

x

xfxxf

x

yxx

)()(limlim

00

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

,dx

dy),(xf

dx

d ,'y ).(' xf

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 = x0 if the derivative of the function with respect to x exists at x = x0.

The value of the derivative of y = f(x) with respect to x at x = x0 is denoted

by or .0xxdx

dy

)(' 0xf

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:

0 0

0

0

0

0 0 0

( ) ( ) sin( ) sin'( ) lim lim

sin cos cos sin h sinlim

sin cos sin cos sinlim

cos 1 sinlim sin cos

cos 1limsin lim lim cos lim

h h

h

h

h

h h h h

f x h f x x h xf x

h hx h x x

hx h x x h

h h

h hx x

h h

hx x

h

0

sin h

h

DERIVS. OF TRIG. FUNCTIONS Equation 1

• Two of these four limits are easy to evaluate.

DERIVS. OF TRIG. FUNCTIONS

0 0 0 0

cos 1 sinlimsin lim lim cos limh h h h

h hx x

h h

• Since we regard x as a constant

when computing a limit as h → 0,

we have:


limh 0

sin x sin x

limh 0

cos x cos x

• 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:

0

sinlim 1


• We can deduce the value of the remaining

limit in Equation 1 as follows.

0

0

2

0

cos 1lim

cos 1 cos 1lim

cos 1

cos 1lim

(cos 1)


2

0

0

0 0

0

sinlim

(cos 1)

sin sinlim

cos 1

sin sin 0lim lim 1 0

cos 1 1 1

cos 1lim 0


• If we put the limits (2) and (3) in (1),

we get:

• So, we have proved the formula for sine,

0 0 0 0

cos 1 sin'( ) limsin lim lim cos lim

(sin ) 0 (cos ) 1

cos

h h h h

h hf x x x

h hx x

x

DERIVS. OF TRIG. FUNCTIONS Formula 4

(sin ) cosd

x xdx

• Differentiate y = x2 sin x.– Using the Product Rule and Formula 4,

we have:

2 2

2

(sin ) sin ( )

cos 2 sin

dy d dx x x x

dx dx dx

x x x x

Example 1DERIVS. OF TRIG. FUNCTIONS

Figure 3.4.3, p. 151

• Using the same methods as in

the proof of Formula 4, we can prove:

(cos ) sind

x xdx

Formula 5DERIV. OF COSINE FUNCTION

2

2

2 22

2 2

2

sin(tan )

cos

cos (sin ) sin (cos )

coscos cos sin ( sin )

cos

cos sin 1sec

cos cos

(tan ) sec

d d xx

dx dx x

d dx x x x

dx dxx

x x x x

x

x xx

x xd

x xdx

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.

2 2

(sin ) cos (csc ) csc cot

(cos ) sin (sec ) sec tan

(tan ) sec (cot ) csc

d dx x x x x

dx dxd d

x x x x xdx dxd d

x x x xdx dx


• Differentiate

• For what values of x does the graph of f have

a horizontal tangent?

sec( )

1 tan

xf x

x


• The Quotient Rule gives:

2

2

2

2 2

2

2

(1 tan ) (sec ) sec (1 tan )'( )

(1 tan )

(1 tan )sec tan sec sec

(1 tan )

sec (tan tan sec )

(1 tan )

sec (tan 1)

(1 tan )

d dx x x x

dx dxf xx

x x x x x

x

x x x x

x

x x

x

Example 2Solution:

tan2 x + 1 = sec2 x

• Find the 27th derivative of cos x.

– The first few derivatives of f(x) = cos x are as follows:

(4)

(5)

'( ) sin

''( ) cos

'''( ) sin

( ) cos

( ) sin

f x x

f x x

f x x

f x x

f x x


– 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 4Solution:

• Find

– In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7:

0

sin 7lim

4x

x

x

sin 7 7 sin 7

4 4 7

x x

x x


• If we let θ = 7x, then θ → 0 as x → 0.

So, by Equation 2, we have:

0 0

0

sin 7 7 sin 7lim lim

4 4 7

7 sinlim

4

7 71

4 4

x x

x x

x x

Example 5Solution:

• Calculate .

– We divide the numerator and denominator by x:

by the continuity of cosine and Eqn. 2

0lim cotx

x x


0 0 0

0

0

cos coslim cot lim lim

sinsin

lim cos cos0sin 1lim

1

x x x

x

x

x x xx x

xxx

x

x

x

THANK YOU

Limits And Derivative

Education

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