Limit Definition of the Derivative. Objective To use the limit definition to find the derivative of...

Limit Definition of the Limit Definition of the DerivativeDerivative

ObjectiveObjective

To use the limit definition to find the To use the limit definition to find the derivative of a function.derivative of a function.

TS: Devoloping a capacity for working TS: Devoloping a capacity for working within ambiguity.within ambiguity.

SlopeSlope

SlopeSlope: the rate at which a line rises or falls: the rate at which a line rises or falls

For a line, the rate (or slope) is the same For a line, the rate (or slope) is the same at every point on the line.at every point on the line.

For graphs other than lines, the rate at For graphs other than lines, the rate at which the graph rises or falls changes which the graph rises or falls changes from point to point.from point to point.

SlopeSlope

This parabola is rising This parabola is rising more quickly at point more quickly at point A than it is at point B.A than it is at point B.

At the vertex, point C, At the vertex, point C, the graph levels off.the graph levels off.

At point D the graph At point D the graph is falling.is falling.

SlopeSlope

To determine the To determine the raterate at which a graph at which a graph rises or falls at a rises or falls at a singlesingle point, we can find point, we can find the slope of the the slope of the tangenttangent line to the point. line to the point.

How do we calculate the slope of a How do we calculate the slope of a tangent line?tangent line?

Video Clip fromVideo Clip fromCalculus-Help.comCalculus-Help.com

The Difference QuotientThe Difference Quotient


The The derivativederivative is the slope of the is the slope of the tangenttangent line to a graph line to a graph f(x)f(x), and is usually denoted , and is usually denoted f’(x)f’(x)..

To calculate the slope of the To calculate the slope of the tangenttangent line line we will use the we will use the differencedifference quotientquotient..

Limit Definition of the Derivative Limit Definition of the Derivative

The derivative is the The derivative is the formulaformula which gives which gives

the slope of the tangent line at any point the slope of the tangent line at any point xx

for for f f ((xx)), and is denoted, and is denoted

provided this limit exists.provided this limit exists.

0

( ) ( )'( ) lim

x

f x x f xf x

x

DerivativesDerivatives

The derivative of the function The derivative of the function y y = = f f ((xx) ) may be may be expressed as …expressed as …

'( )f x

'y

dy

dx

Prime notation

Leibniz notation

“f prime of x”

“y prime”

“the derivative of y with respect to x”

DerivativesDerivatives

The process of finding derivatives is called The process of finding derivatives is called differentiationdifferentiation..

A function is A function is differentiabledifferentiable at a point if its at a point if its derivativederivative exists at that point. exists at that point.

Limit Definition of the DerivativeLimit Definition of the Derivative

Use the limit definition to find the Use the limit definition to find the derivative of:derivative of:

2( ) 3 5f x x x

0

( ) ( )'( ) lim

x

f x x f xf x

x

Limit Definition of the DerivativeLimit Definition of the Derivative2( ) 3 5f x x x

0

( ) ( )'( ) lim

x

f x x f xf x

x

2 2

0

( ) 3( ) 5 ( 3 5)'( ) lim

x

x x x x x xf x

x

2 2 2

0

2 ( ) 3 3 5 3 5'( ) lim

x

x x x x x x x xf x

x


0

(2 3)'( ) lim

x

x x xf x

x

2

0

2 ( ) 3'( ) lim

x

x x x xf x

x

'( ) 2 3f x x

A formula for finding theslope of the tangent lineof f (x) at a given point.


Use the limit definition to find the Use the limit definition to find the derivative of:derivative of:

2( ) 8 1f x x

0

( ) ( )'( ) lim

x

f x x f xf x

x

Limit Definition of the DerivativeLimit Definition of the Derivative2( ) 8 1f x x

0

( ) ( )'( ) lim

x

f x x f xf x

x

2 2

0

8( ) 1 (8 1)'( ) lim

x

x x xf x

x

2 2 2

0

8( 2 ( ) ) 1 8 1'( ) lim

x

x x x x xf x

x


0

(16 8 )'( ) lim

x

x x xf x

x

'( ) 16f x x

A formula for finding theslope of the tangent lineof f (x) at a given point.

2 2 2

0

8 16 8( ) 1 8 1'( ) lim

x

x x x x xf x

x

DifferentiabilityDifferentiability

Not every function is differentiable at all Not every function is differentiable at all points. points.

Some common situations in which a Some common situations in which a function will not be differentiable at a point function will not be differentiable at a point include:include:1. Vertical tangent lines1. Vertical tangent lines2. Discontinuities (like a hole, break, or vertical 2. Discontinuities (like a hole, break, or vertical

asymptote)asymptote)3. Sharp turns (called cusps & nodes)3. Sharp turns (called cusps & nodes)

DifferentiabilityDifferentiability

CALCULUS JEOPARDY!CALCULUS JEOPARDY!

$200$200

Answer:Answer:

It’s computed by finding the limit of the It’s computed by finding the limit of the difference quotient as ∆x approaches 0.difference quotient as ∆x approaches 0.

Question:Question:

What is the derivative?What is the derivative?


$400$400

Answer:Answer:

It’s used to find the slope of a It’s used to find the slope of a function at a point. function at a point.

Question:Question:



$600$600

Answer:Answer:

It’s used to find the slope of the tangent It’s used to find the slope of the tangent line to a graph line to a graph f f ((xx)), and is usually denoted , and is usually denoted

f’f’((xx)). .

Question:Question:



$800$800

Answer:Answer:

It’s used to find the instantaneous rate of It’s used to find the instantaneous rate of change of a function. change of a function.

Question:Question:



$1000$1000

Answer:Answer:

It’s the thing we love most about calculus. It’s the thing we love most about calculus.

Question:Question:


The The DerivativeDerivative is… is…

computed by finding the computed by finding the limitlimit of the difference of the difference quotient as ∆x approaches 0.quotient as ∆x approaches 0.

the the slopeslope of a function at a point. of a function at a point.

the slope of the the slope of the tangenttangent line to a graph line to a graph f f ((xx)), and , and is usually denoted is usually denoted f’f’((xx))..

the the instantaneousinstantaneous rate of change of a function. rate of change of a function.

Limit Definition of the Derivative. Objective To use the limit definition to find the derivative of...

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