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)


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

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


2. dx

dv

dx

duvu

dx

d )(


3. dx

duccu

dx

d)(


4. 1)( nn nxxdx

d for any positive integer n


4. 1)( nn nxxdx

d for any positive integer n

Binominal Theorem


5. dx

duv

dx

dvuuv

dx

d)( product rule


6.

2)(

vdxdv

udxdu

v

v

u

dx

d

where v ≠ 0

quotient rule


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

Limits and derivatives

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