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2.1 Limit definition of the Derivative and Differentiability 2015

Devil’s Tower, WyomingGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

Warm-Up

Find the following limit (without a calculator):

1

2lim

1x

x

x

HWQ

Find the following limit (without a calculator):

1lim cosx x

1

The slope of a line is given by:y

mx

x

y

The slope at (1,1) can be approximated by the slope of the secant through (4,16).

y

x

16 1

4 1

15

3 5

We could get a better approximation if we move the point closer to (1,1). ie: (3,9)

y

x

9 1

3 1

8

2 4

Even better would be the point (2,4).

y

x

4 1

2 1

3

1 3

2f x x

The slope of a line is given by:y

mx

x

y

If we got really close to (1,1), say (1.1,1.21), the approximation would get better still

y

x

1.21 1

1.1 1

.21

.1 2.1

How far can we go?

2f x x

1f

1 1 h

1f h

h

slopey

x

1 1f h f

h

slope at 1,1 2

0

1 1limh

h

h

2

0

1 2 1limh

h h

h

0

2limh

h h

h

2

The slope of the curve at any point is: y f x ,P x f x

0

lim h

f x h f xm

h

Differentiability and Continuity

2f x x

The slope of the curve at any point is: y f x ,P x f x

0

lim h

f x h f xm

h

Slope at any point on the graph of a function:

Slope at a specific point on the graph of a function:

The slope of the curve at the point is: y f x ,P c f c

lim x c

f x f cf c

x c

2 1f x x

Using the limit of the difference quotient, find the slope of the line tangent to the graph of the given function at x= -1, then use the slope to find the equation of the tangent line:

Example:

2 2 1y x

Try this one:

2 ,

Use the limit definition of the

derivative to find 2

f x x

f

12

2f

Another Example ?:

132 2 xxxf

Find the equation of a line tangent to the graph at (-2,-1)

Differentiability and Continuity

The following statements summarize the relationship

between continuity and differentiability.

1. If a function is differentiable at x = c, then it is continuous at x = c. So, differentiability implies continuity.

2. It is possible for a function to be continuous at x = c and not be differentiable at x = c. So, continuity does not imply differentiability.

1. If a function is differentiable at x = c, then it is continuous at x = c. Differentiability implies continuity.

2. It is possible for a function to be continuous at x = c and not differentiable at x = c. So, continuity does not imply differentiability.

3. Continuous functions that have sharp turns, corner points or cusps, or vertical tangents are not differentiable

at that point.

Very Important, so we’ll say it again:

Differentiability and Continuity

The following alternative limit form of the derivative is useful in investigating the relationship between differentiability and continuity. The derivative of f at c is

provided this limit exists (see Figure 2.10).

Figure 2.10

Differentiability and Continuity

Note that the existence of the limit in this alternative formrequires that the one-sided limits

exist and are equal.

These one-sided limits are called the derivatives from the left and from the right, respectively.

It follows that f is differentiable on the closed interval [a, b] if it is differentiable on (a, b) and if the derivative from the right at a and the derivative from the left at b both exist.

Figure 2.11

Differentiability and Continuity

If a function is not continuous at x = c, it is also not differentiable at x = c.

For instance, the greatest integer function is not continuous at x = 0, and so it is not differentiable at x = 0 (see Figure 2.11).

The function shown in Figure 2.12 is continuous at x = 2.

Example of a function not differentiable at every point – A Graph with a Sharp Turn

Figure 2.12

However, the one-sided limits

and

are not equal.

So, f is not differentiable at x = 2 and the graph of f does not have a tangent line at the point (2, 0).

cont’d

Example of a function not differentiable at every point –A Graph with a Sharp Turn

For example, the function shown in Figure 2.7 has a vertical tangent line at (c, f(c)).

Figure 2.7

Example of a function not differentiable at c.

Example:

1

3f x x

Determine whether the function is continuous at x=0. Is it differentiable there? Use to analyze the

derivative at x=0.

limx c

f x f c

x c

Not differentiable at x=0Vertical tangent line

1 3

0 0

0 0lim lim

0x x

f x f x

x x

2 30

1limx x

Example:

1

1 22

2 2

x xf x

x x

Determine whether the function is differentiable at x = 2.

1)f(x) is continuous at x=2 and 2)The left hand and right hand derivatives agree

Differentiable at x = 2 because:

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.

A function will not have a derivative1)Where it is discontinuous 2)Where it has a sharp turn 3)Where it has a vertical tangent

Recap: To be differentiable, a function must be continuous and smooth.

Derivatives will fail to exist at:

corner cusp

vertical tangent discontinuity

f x x 2

3f x x

3f x x 1, 0

1, 0

xf x

x

y f x

y f x

The derivative is the slope of the original function.

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

2 3y x

2 2

0

3 3limh

x h xy

h

2 2 2

0

2limh

x xh h xy

h

2y x

0lim 2h

y x h

0

Homework

• P. 103 8,17,23,25,30,35-47 odd, 81-87 odd