The Derivative and the Tangent Line Problem

1 / 54

# The Derivative and the Tangent Line Problem - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'The Derivative and the Tangent Line Problem' - ikia

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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
• understand the relationship between differentiability and continuity
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

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 ofa Tangent
• Let Δx shrinkfrom the left
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

Let

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

Note:

dx does not mean d times x !

dy does not mean d times y !

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.)

does not mean times !

Note:

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

The derivative is the slope of the original function.

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

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

Example: Find the slope of the graph of

at the point (2, 5).

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 Line

Example: Find the slope of the graph of

at the point (2, 5).

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 Line

Example: Find the slope of the graph of

at the point (2, 5).

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

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 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 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 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 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 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 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 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 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 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 Function

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

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

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

Derivative

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

Derivative

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

Derivative

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

Derivative

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

Derivative

Example: Find for

Derivative

Example: Find for

Derivative

Example: Find for

Derivative

Example: Find for

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

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-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

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)
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.
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.

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
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.

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

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

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
• 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 !