limits and derivatives

Limits and Derivatives

Concept of a Function

FUNCTIONS

• “FUNCTION” indicates a relationship among objects.

• A FUNCTION provides a model to describe a system.

• A FUNCTION expresses the relationship of one variable or a group of variables (called the domain) with another variables( called the range) by associating every member in the domain to a unique member in range.

TYPES OF FUNCTIONS

• LINEAR FUNCTIONS

• INVERSE FUNCTIONS

• EXPONENTIAL FUNCTIONS

• LOGARITHMIC FUNCTIONS

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

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.

Differentiation Rules

1. 0)( cdx

d

Differentiation Rules

1. 0)( cdx

d

Differentiation Rules

2. dx

dv

dx

duvu

dx

d )(

Differentiation Rules

2. dx

dv

dx

duvu

dx

d )(

Differentiation Rules

2. dx

dv

dx

duvu

dx

d )(

Differentiation Rules

3. dx

duccu

dx

d)(

Differentiation Rules

3. dx

duccu

dx

d)(

Differentiation Rules

4. 1)( nn nxxdx

d for any positive integer n

Differentiation Rules

4. 1)( nn nxxdx

d for any positive integer n

Binominal Theorem

Differentiation Rules

5. dx

duv

dx

dvuuv

dx

d)( product rule

Differentiation Rules

5. dx

duv

dx

dvuuv

dx

d)( product rule

Differentiation Rules

6.

2)(

vdxdv

udxdu

v

v

u

dx

d

where v ≠ 0

quotient rule

Differentiation Rules

6.

2)(

vdxdv

udxdu

v

v

u

dx

d

where v ≠ 0

quotient rule

Differentiation Rules

7. 1)( nn nxxdx

d for any integer n

DIFFERENTIATION RULES

• y,u and v are functions of x. a,b,c, and n are constants (numbers).

The derivative of a constant is zero. Duh! If everything is constant, that means its rate, its derivative, will be zero. The graph of a constant, a number is a horizontal line. y=c. The slope is zero.

The derivative of x is 1. Yes. The graph of x is a line. The slope of y = x is 1. If the graph of y = cx, then the slope, the derivative is c.

1xdx

d

0cdx

d

MORE RULES

• When you take the derivative of x raised to a power (integer or fractional), you multiply expression by the exponent and subtract one from the exponent to form the new exponent.

1 nn nxxdx

d

23 3xxdx

d

OPERATIONS OF DERIVATIVES• The derivative of the sum or

difference of the functions is merely the derivative of the first plus/minus the derivative of the second.

dx

duv

dx

dvuuv

dx

d

dx

dv

dx

duvu

dx

d

• The derivative of a product is simply the first times the derivative of the second plus second times the derivative of the first.

2vdxdv

udxdu

v

v

u

dx

d

• The derivative of a quotient is the bottom times the derivative of the top, minus top times the derivative of the bottom….. All over bottom square..

• TRICK: LO-DEHI – HI-DELO

• LO2

JUST GENERAL RULES

• If you have constant multiplying a function, then the derivative is the constant times the derivative. See example below:

• The coefficient of the x6 term is 5 (original constant) times 7 (power rule.)

67 355 xxdx

d

dx

dvccv

dx

d

SECOND DERIVATIVES

• You can take derivatives of the derivative. Given function f(x), the first derivative is f’(x). The second derivative is f’’(x), and so on and so forth.

• Using Leibniz notation of dy/dx

2

2

dx

yd

dx

dy

dx

d

For math ponders, if you are interesting in the Leibniz notation of derivatives further, please see my article on that. Thank you. Hare Krishna >=) –Krsna Dhenu

EXAMPLE 4:

• Find the derivative:

• Use the power rule and the rule of adding derivatives.

• Note 3/2 – 1 = ½. x½ is the square root of x.

• Easy eh??

22

35 22 xxxy

xxxy 435 4

EXAMPLE 5

• Find the equation of the line tangent to y = x3 +5x2 –x + 3 at x=0.

• First find the (x,y) coordinates when x = 0. When you plug 0 in for x, you will see that y = 3. (0,3) is the point at x=0.

• Now, get the derivative of the function. Notice how the power rule works. Notice the addition and subtraction of derivative. Notice that the derivative of x is 1, and the derivative of 3, a constant, is zero.

35 23 xxxy

1103 2 xxdx

dy

EX 5 (continued)

• Now find the slope at x=0, by plugging in 0 for the x in the derivative expression. The slope is -1 since f’(0) = -1.

• Now apply it to the equation of a line.

10

xdx

dy

)( 00 xxmyy

EX 5. (continued)

• Now, plug the x and y coordinate for x0 and y0 respectively. Plug the slope found in for m.

• And simplify

• On the AP, you can leave your answer as the first form. (point-slope form)

)0(13 xy

3 xy

EXAMPLE 6

• Find all the derivatives of y = 8x5.

• Just use the power rule over and over again until you get the derivative to be zero.

• See how the power rule and derivative notation works?

0

960

960

480

160

40

8

6

6

5

5

4

4

23

3

32

2

4

5

dx

yd

dx

yd

xdx

yd

xdx

yd

xdx

yd

xdx

dy

xy