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### The Derivative and the Tangent Line Problem

The Slope of the Graph of a Non-Linear FunctionThe Slope of the Graph of a Non-Linear FunctionThe Slope of the Graph of a Non-Linear Function

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.

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.)

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 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:

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

a) x = 1

b) x = -2

a) x = 1:

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

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

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.

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

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 !

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