Lesson 2.1Lesson 2.1The Tangent Line The Tangent Line ProblemProblemByByDarren DrakeDarren Drake05/16/0605/16/06

HistoryHistory

Calculus grew from 4 major problemsCalculus grew from 4 major problems Velocity/acceleration problemVelocity/acceleration problem Max and min value problemMax and min value problem Area problemArea problem THE TANGENT LNE PROBLEMTHE TANGENT LNE PROBLEM

HistoryHistory

Pierre de Format, Rene’ Descartes, Pierre de Format, Rene’ Descartes, Christian Huygens, and Isaac Barrow are Christian Huygens, and Isaac Barrow are given credit for finding partial solutionsgiven credit for finding partial solutions

Isaac Newton(1642-1727) is credited for Isaac Newton(1642-1727) is credited for finding the general solution to the tangent finding the general solution to the tangent line problemline problem

What is a Tangent Line?What is a Tangent Line?

The Tangent line touches the curve at The Tangent line touches the curve at one pointone point

But this doesn’t work for all curvesBut this doesn’t work for all curves

How do you find the How do you find the tangent line?tangent line?

To find the tangent line at point c, you must To find the tangent line at point c, you must first find the slope of the tangent line at point cfirst find the slope of the tangent line at point c

You can approximate this slope by of the You can approximate this slope by of the secant line containing the points andsecant line containing the points and

slope of the secant =slope of the secant =

Recall Recall

( , ( )c x f c x ( , ( ))c f c

2 1

2 1

y ym

x x

( ) ( ) ( ) ( )

( )

f c x f c f c x f c

c x c x

The closer is to 0, the closer is to cThe closer is to 0, the closer is to c The more accurate the tangent line The more accurate the tangent line

approximation will beapproximation will be

Hmmmm….Hmmmm….

this sounds like a limit!!!this sounds like a limit!!!

x c x

Soo…Soo…

If If ff is defined on an openinterval containing is defined on an openinterval containing c, and if the limit c, and if the limit

Exists, then the line passing through the Exists, then the line passing through the point point (c(c, , ff (c))(c)) with the slope with the slope m m is the line is the line tangent to the graph tangent to the graph f(x) f(x) at the point at the point (c(c, , ff (c))(c))

0

( ) ( )lim limx x

y f c x f cm

x x

Example 1Example 1The slope of the graph of a The slope of the graph of a linear functionlinear function

Given Given f(x) = 2x-3, f(x) = 2x-3, find the slope atfind the slope at (2,1) (2,1)

0 0

(2 ) [2(2 ) 3] [2(2) 3]lim limx x

f x x

x

0

0

0

4 2 3 4 3lim

2lim

lim 2

2

x

x

x

x

xx

x

Example 2Example 2Tangent Lines to the graph Tangent Lines to the graph of a nonlinear functionof a nonlinear function

Find the slope at Find the slope at (0,1)(0,1) of the tangent line to of the tangent line to the graph of and write an equation the graph of and write an equation for the tangent line at this point for the tangent line at this point

2( ) 1f x x

2 2

0

( ) 1 ( 1)limx

x x x

x

2 2 2

0

2

0

0

2 ( ) ( ) 1 1lim

2 ( ) ( )lim

lim (2 )

2

x

x

x

x x x x x

x

x x x

xx x

x

1 1( )

1 2(0)( 0)

1 0

1

y y m x x

y x

y

y

Example 3Example 3Finding the derivative by Finding the derivative by the limit processthe limit process

The limit used to define the slope of the tangent line is also used to The limit used to define the slope of the tangent line is also used to

define differentiation. The derivative of define differentiation. The derivative of f f atat x x is given byis given by

or or

Find the derivative of Find the derivative of

0

( ) ( )( ) lim

x

f x x f xf x

x

3( ) 2f x x x 3 2 2 3 3

0

2 2 2

0

2 2

0

2 2

0

2

3 3 ( ) ( ) 2 2lim

3 3 ( ) ( ) 2lim

[3 3 ( ) 2lim

lim[3 3 ( ) 2]

3 2

x

x

x

x

x x x x x x x x x

x

x x x x x

x

x x x x x

x

x x x x

x

3 3

0

( ) 2( ) ( 2 )( ) lim

x

x x x x x xf x

x

( ) ( )( ) lim

x c

f x f cf c

x c

Example 4Example 4DifferentiabliltyDifferentiablilty

Derivatives haveDerivatives have

certain rules on certain rules on

when they exist when they exist

ContinuityContinuity

Difeferentiability Difeferentiability

Vertical tangentVertical tangent

Example 5Example 5ApplicationsApplications

There is a hill whose cross There is a hill whose cross section forms the equation section forms the equation

. Your car just died so . Your car just died so you have to push your car up the you have to push your car up the hill. But the hill is too steep at hill. But the hill is too steep at first. So You make a ramp to first. So You make a ramp to make it up the hill but you don’t make it up the hill but you don’t know how steep it is. You won’t know how steep it is. You won’t be able to push your car up the be able to push your car up the ramp if it is steeper than 1/3 ramp if it is steeper than 1/3 When the ramp is at your feet, it When the ramp is at your feet, it touches the hill at one point 4 touches the hill at one point 4 feet from the start of the hill. How feet from the start of the hill. How steep is the ramp? Will you be steep is the ramp? Will you be able to push you car up it? Use able to push you car up it? Use the definition of the tangent line the definition of the tangent line to find your answerto find your answer

( )f x x

0

0

0

0

0

( ) lim

)lim

( )lim

lim

1lim

1

2

x

x

x

x

x

x x xf x

x

x x x x x x

x x x x

x x x

x x x x

x

x x x x

x x x

x

1

2 (4)

1

4

m

m

Example 6Example 6ApplicationApplication

A see saw sits on the A see saw sits on the pivot formed by the pivot formed by the equationequationWhat is the slope of theWhat is the slope of theof the see saw at of the see saw at x=0x=0? ? Use the definition of the Use the definition of the Derivative to find Derivative to find your answeryour answer

RecallRecall

( ) 3f x x

( ) ( )( ) lim

x c

f x f cf c

x c

( ) 3f x x

0

( ) (0)(0) lim

0x

f x ff

x

0

3 3lim 1

0x

x

x

and

0

3 3lim 1

0x

x

x

Deriv. from left

Deriv. from right

The derivative from the left and right do not equal eachother; thereofre, f is not differentiable at x = 0 so we don’t know what the slope of the see saw is at x = 0

Trick Question!!!!!!Trick Question!!!!!!

The slope of the see saw The slope of the see saw is indeterminate, we is indeterminate, we don’t what it is!!don’t what it is!!

The EndThe End