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The Derivative and the Tangent Line Problem. Section 2.1. After this lesson, you should be able to:. find the slope of the tangent line to a curve at a point use the limit definition of a derivative to find the derivative of a function

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after this lesson you should be able to
After this lesson, you should be able to:
  • find the slope of the tangent line to a curve at a point
  • use the limit definition of a derivative to find the derivative of a function
  • understand the relationship between differentiability and continuity
tangent line
Tangent Line

A line is tangent to a curve at a point P if the line is perpendicular to the radial line at point P.

Note: Although tangent lines do not intersect a circle, they may cross through point P on a curve, depending on the curve.

P

the tangent line problem
The Tangent Line Problem

Find a tangent line to the graph of f at P.

Why would we want a tangent line???

f

Remember, the closer you zoom in on point P, the more the graph of the function and the tangent line at P resemble each other. Since finding the slope of a line is easier than a curve, we like to use the slope of the tangent line to describe the slope of a curve at a point since they are the same at a particular point.

A tangent line at P shares the same point and slope as point P. To write an equation of any line, you just need a point and a slope. Since you already have the point P, you only need to find the slope.

P

definition of a tangent
Definition ofa Tangent
  • Let Δx shrinkfrom the left
the derivative of a function
The Derivative of a Function

Differentiation- the limit process is used to define the slope of a tangent line.

Really a fancy slope formula… change in y divided by the change in x.

Definition of Derivative: (provided the limit exists,)

This is a major part of calculus and we will differentiate until the cows come home!

Also,

= slope of the line tangent to the graph of f at (x, f(x)).

= instantaneous rate of change of f(x) with respect to x.

notations for derivative
Notations For Derivative

Let

If the limit exists at x, then we say that f is differentiable at x.

slide10
Note:

dx does not mean d times x !

dy does not mean d times y !

slide11
does not mean !

does not mean !

Note:

(except when it is convenient to think of it as division.)

(except when it is convenient to think of it as division.)

slide12
does not mean times !

Note:

(except when it is convenient to treat it that way.)

slide13
The derivative is the slope of the original function.

The derivative is defined at the end points of a function on a closed interval.

slide15
A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

p

the slope of the graph of a line
The Slope of the Graph of a Line

Example: Find the slope of the graph of

at the point (2, 5).

the slope of the graph of a line18
The Slope of the Graph of a Line

Example: Find the slope of the graph of

at the point (2, 5).

the slope of the graph of a line19
The Slope of the Graph of a Line

Example: Find the slope of the graph of

at the point (2, 5).

the slope of the graph of a line20
The Slope of the Graph of a Line

Example: Find the slope of the graph of

at the point (2, 5).

the slope of the graph of a line21
The Slope of the Graph of a Line

Example: Find the slope of the graph of

at the point (2, 5).

the slope of the graph of a line22
The Slope of the Graph of a Line

Example: Find the slope of the graph of

at the point (2, 5).

the slope of the graph of a non linear function
The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent lines at:

a) x = 1

b) x = -2

a) x = 1:

the slope of the graph of a non linear function24
The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent lines at:

a) x = 1

b) x = -2

a) x = 1:

the slope of the graph of a non linear function25
The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent lines at:

a) x = 1

b) x = -2

a) x = 1:

the slope of the graph of a non linear function26
The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent lines at:

a) x = 1

b) x = -2

a) x = 1:

the slope of the graph of a non linear function27
The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent lines at:

a) x = 1

b) x = -2

a) x = 1:

the slope of the graph of a non linear function28
The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent lines at:

a) x = 1

a) x = 1:

b) x = -2

the slope of the graph of a non linear function29
The Slope of the Graph of a Non-Linear Function

Example: Given , find f ’(x) and the equation of the tangent line at:

b) x = -2

the slope of the graph of a non linear function30
The Slope of the Graph of a Non-Linear Function

Example: Find f ’(x) and the equation of the tangent line at x = 2 if

the slope of the graph of a non linear function31
The Slope of the Graph of a Non-Linear Function

Example: Find f ’(x) and the equation of the tangent line at x = 2 if

the slope of the graph of a non linear function32
The Slope of the Graph of a Non-Linear Function

Example: Find f ’(x) and the equation of the tangent line at x = 2 if

the slope of the graph of a non linear function33
The Slope of the Graph of a Non-Linear Function

Example: Find f ’(x) and the equation of the tangent line at x = 2 if

example continued
Example-Continued

If x = 2, the slope is, -¼. So, y = 1/4x + b. Going back to the original equation of y = 1/x, we see if x = 2, y = 1/2. So:

derivative
Derivative

Example: Find the derivative of f(x) = 2x3 – 3x.

derivative36
Derivative

Example: Find the derivative of f(x) = 2x3 – 3x.

derivative37
Derivative

Example: Find the derivative of f(x) = 2x3 – 3x.

derivative38
Derivative

Example: Find the derivative of f(x) = 2x3 – 3x.

derivative39
Derivative

Example: Find the derivative of f(x) = 2x3 – 3x.

derivative40
Derivative

Example: Find for

derivative41
Derivative

Example: Find for

derivative42
Derivative

Example: Find for

derivative43
Derivative

Example: Find for

THIS IS A HUGE RULE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

example continued44
Example-Continued

Let’s work a little more with this example…

Find the slope of the graph of f at the points (1, 1) and (4, 2). What happens at (0, 0)?

example continued45
Example-Continued

Let’s graph tangent lines with our calculator…we’ll draw the tangent line at x = 1.

Graph the function on your calculator.

1

3

Select 5: Tangent(

Type the x value, which in this case is 1, and then hit 

4

(I changed my window)

2

Now, hit  DRAW

Here’s the equation of the tangent line…notice the slope…it’s approximately what we found

differentiability implies continuity
Differentiability Implies Continuity

If f is differentiable at x, then f is continuous at x.

  • Some things which destroy differentiability:
  • A discontinuity (a hole or break or asymptote)
  • A sharp corner (ex. f(x)= |x| when x = 0)
  • A vertical tangent line (ex: when x = 0)
slide47
2.1 Differentiation Using Limits of Difference Quotients
  • Where a Function is Not Differentiable:
  • 1) A function f(x) is not differentiable at a point x = a, if there is a “corner” at a.
slide48
2.1 Differentiation Using Limits of Difference Quotients
  • Where a Function is Not Differentiable:
  • 2) A function f (x) is not differentiable at a point
  • x = a, if there is a vertical tangent at a.
slide49
3. Find the slope of the tangent line to at x = 2.

This function has a sharp turn at x = 2.

Therefore the slope of the tangent line at x = 2 does not exist.

  • Functions are not differentiable at
  • Discontinuities
  • Sharp turns
  • Vertical tangents
slide50
2.1 Differentiation Using Limits of Difference Quotients

Where a Function is Not Differentiable:

3) A function f(x) is not differentiable at a point x = a, if it is not continuous at a.

Example: g(x) is not

continuous at –2,

so g(x) is not

differentiable at x = –2.

slide51
4. Find any values where is not differentiable.

This function has a V.A. at x = 3.

Therefore the derivative at x = 3 does not exist.

Theorem:

If f is differentiable at x = c,

then it must also be continuous at x = c.

example
Example

Find an equation of the line that is tangent to the graph of f and parallel to the given line.

f(x) = x3 + 2 Line: 3x – y – 4 = 0

example53
Example

Find an equation of the line that is tangent to the graph of f and parallel to the given line.

f(x) = x3 + 2 Line: 3x – y – 4 = 0

Taking my word for it, the derivative of the function is

This is 3 when x is

definition of derivative
Definition of Derivative
  • The derivative is the formula which gives the slope of the tangent line at any point x for f(x)
  • Note: the limit must exist
    • no hole
    • no jump
    • no pole
    • no sharp corner

A derivative is a limit !

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