Done BY, Achuthan xi b k.v.pattom

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)

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) 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 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

Consider the function

The Idea of Limits

x

xxh )(

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

g(x)

Consider the function

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

Theorems On Limits

Theorems On Limits

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

Theorems of Limits at Infinity

Theorems of Limits at Infinity

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:

DERIVS. OF TRIG. FUNCTIONS

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

DERIVS. OF TRIG. FUNCTIONS Equation 2

• 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)

DERIVS. OF TRIG. FUNCTIONS

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

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,

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

DERIVS. OF TRIG. FUNCTIONS

• Differentiate

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

a horizontal tangent?

sec( )

1 tan

xf x

x

Example 2DERIVS. OF TRIG. FUNCTIONS

• 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

Example 4DERIVS. 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 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

Example 5DERIVS. OF TRIG. FUNCTIONS

• 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

Example 6DERIVS. OF TRIG. FUNCTIONS

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

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