ch 2a derivative by definition

WS Chapter 2 Derivative 1

CHAPTER 2

DERIVATIVE

Jan 2012

2

DERIVATIVE BY DEFINITIONLEARNING OBJECTIVES :

At the end of the module , you should be able to :

Compute derivative by using definition

Jan 2012 WS Chapter 2 Derivative


INTRODUCTON

The Delta Notation

Let f be a function.

Let x be the independent variable

And y be the corresponding value of f.

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Let :

represent a small change in the value of x fromThe corresponding small change in the value of y

is the average of rate of change of f between

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THE DERIVATIVE

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The limit,


A function is said to be differentiable at a point If the derivative of the function exists at that point.

If a function is differentiable, then the function must be continuous.

However, the converse is not true.

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Suppose that a particle in rectilinear motion along an s-axis has position function s = f(t)

Therefore the average velocity of the particle over a time interval is defined to be

If is a given time interval

Then, we define the displacement or change in position of the particle over this time interval to be the difference between

its final and initial position coordinates

Displacement over the interval

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What is instantaneous velocity?

For particle in rectilinear motion Average velocity – behavior over an interval time

Instantaneous velocity – behavior at a specific instant time

We define the instantaneous velocity of the particle at time to be the limit as of its average velocity over time intervals between

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Example 1

Suppose that an object is released from rest (i.e. initial velocity is zero) from a building from a height of 1250 ft above street level. It is shown in physics with appropriate assumptions, the objects height s above the street level, t seconds after its release , can be modeled by

Verify that the object has not reached the ground at t = 5s, and find its instantaneous velocity at that time.

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SOLUTION

When t = 5s, This shows that the object is still falling 5 s after it is released.

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Another examples of application of rate of change are

An engineer might be interested in the rate at which the length of a metal rod changes with temperature.

An economist that might be interested in the rate at which production cost changes with the quantity of a product manufactured.

A microbiologist might be interested in the rate at which the number of bacteria in a colony changes with time.

To find the changes of height of water when the water spilled out from the tank at certain speed

etc

Slopes and Rate of ChangeVelocity can be viewed as rate of change i.e the rate of change of position with respect to time.

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We can define :

Average rate of change of y with respect to x on

Instantaneous rate of change of y with respect to x at to be

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Geometrically, the average rate of change over the intervalis the slope of the secant line through the pointsand

and the inst. rate of change is the slope of the tangent line at the point (since it is the limit of the slopes of the secant line through P)

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Therefore, if we let

Average rate of change

Instantaneous rate of change

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WS Chapter 2 Derivative 15Jan 2012

Some examples of instantaneous rate of change are Slope

Velocity Acceleration

Volume Area etc


Example

a) Find the average rate of change of f with respect to x over the interval [2, 6]

b) Find the instantaneous rate of change of f with respect to x when x = -2


SOLUTION


Derivative as Tangent or Slope to a Curve

DefinitionThe tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope

, provided the limit exists.

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Example 1 Find an equation of the tangent line to the parabola

SOLUTION

Using the point slope form of the equation of a line,

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There is an alternative way of expressing the slope of a tangent line that is commonly used.

so the slope of the secant line PQ is

Now the expression for the slope of the tangent line has becomes

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Example 2 Find an equation for the tangent line to the curve

SOLUTION

The point corresponding to x=2 is (2, 1), since f(2) = 1

An equation of the tangent line is

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WS Chapter 2 Derivative – by definition 24Jan 2012

Derivative of a function at x = a

The derivative of a function f at a number x = a is denoted by f ‘ (a) , is defined as follows

OR


Example 1

SOLUTION

Find the derivative of the function

By (1)


SOLUTION By (2)


Derivative of f as a function

The derivative of a function f with respect to variable x is denoted by f ‘ (x) and is defined by

Also known as First Principle Method


Example 2

SOLUTION


Example 3

SOLUTION

Is there any difference between the domain of


Example 4

SOLUTION

What is the difference between the domain of


Example 5

SOLUTION


Other notations for derivative

If , then the derivative can also be denoted as


DEFINITION

A function f is differentiable at “a” if f ’(a) exists

It is differentiable on an open interval ( a , b )

if it is differentiable at every number in the interval


Example 6

SOLUTION


We have to investigate whether

exist ?

Thus f is differentiable at all x except 0.


Both continuity and differentiability are desirable properties for a function to have.

Theorem

If f is differentiable at x = a, then f is continuous at x = a

** However, the converse is not true. (Refer to ex. 6)


3 possibilities a function is not differentiable

1.If the graph of the function has a “sharp corner” in it, the graph has no tangent at this point and therefore the function is not differentiable at that point.

y

x0


2.If the graph of the function has jump discontinuity (i.e the function is discontinuous), then the function is not differentiable.

y

x0


3.If the graph of the function f has vertical tangent line at x = a, that is, f is continuous at x = a and

** This means that the tangent line becomes steeper and steeper as

y

x0 a

ch 2a derivative by definition

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