3.1

Derivative of a Function

Quick Review

h

hh

42lim .1

2

0

2

3lim .2

2

xx

4

1 8

In Exercises 1 – 4, evaluate the indicated limit algebraically.

y

yy 0_lim .3 2

82lim .4

4

x

xx

2

5

Quick Reviewparabola theo tangent tline theof slope theFind .5

0

0 , ,2 and

,23 ofgraph thegconsiderinBy .6 23 xxxf

vertex.itsat 12 xy

.increasing is on which intervals thefind f

Quick Review

1 ,11 ,2

2 xxxx

xf

xfxfxx 11lim and lim Find .7

3

0

In Exercises 7 – 10, let

hfh

1lim Find .80

0

Explain. exist? lim Does .91

xfx

No

different. are limits sided-one twoThe

Explain. ?continuous Is .10 f No1at ousdiscontinu is xf

e.exist thernot doeslimit thebecause

What you’ll learn about Definition of a Derivative Notation Relationship between the Graphs of f and f ' Graphing the Derivative from Data One-sided Derivatives

Essential QuestionWhat is the definition of the derivative and how does it

help us on a graph of a curve at a point?

Definition of Derivative

0

The of the function with respect to the variable is the

function whose value at is

lim

provided the limit exists.

h

f x

f x x

f x h f xf x

h

derivative

The domain of , the set of points in the domain of for which the limit

exists, may be smaller than the domain of . If exists, we say that

at . A

f f

f f x f

x

has a derivative (is differentiable) function that is differentiable

at every point in its domain is a .differentiable function

Differentiable Function

Example Definition of Derivative .3 ateDifferenti 1. 2xxf

h

xhxh

22

0

33lim

h

xh

3lim

2

0

hxh

36lim0

xh6 23h 23x h hx 36

x6

99 ax

Derivative at a Point (alternate)

The derivative of the function at the point where is the limit

lim

provided the limit exists.

x a

f x a

f x f af a

x a

.definition alternate theusing 3 ateDifferenti 2. xxf

ax

axax

33lim

ax

ax

33

33

axax 33

9lim

9 ax

axax 33

a2

3

a6

9

NotationThere are many ways to denote the derivative of a function ( ).

Besides '( ), the most common notations are:

" prime" Nice and brief, but does not name

y f x

f x

y y

the independent variable.

" " or "the derivative Names both variables andof with respect to " uses for derivative.

" " or "the d

dydy dx

dx y x d

dfdf dx

dx

erivative Emphasizes the function's name.of with respect to "

" of at " or "the Emphasis that differentiationderivative of at " is an operation performed on .

f x

df x d dx f x

dx f x f

Relationships between the Graphs of f and f’

Because we can think of the derivative at a point in graphical terms as slope, we can get a good idea of what the graph of the function f’ looks like by estimating the slopes at various points along the graph of f.

We estimate the slope of the graph of f in y-units per x-unit at frequent intervals. We then plot the estimates in a coordinate plane with the horizontal axis in x-units and the vertical axis in slope units.

Graphing the Derivative from DataDiscrete points plotted from sets of data do not yield a continuous curve, but we have seen that the shape and pattern of the graphed points (called a scatter plot) can be meaningful nonetheless. It is often possible to fit a curve to the points using regression techniques. If the fit is good, we could use the curve to get a graph of the derivative visually. However, it is also possible to get a scatter plot of the derivative numerically, directly from the data, by computing the slopes between successive points.

Graphing f ʹ from f3. Use the graph of f (x) to graph the derivative f ʹ(x) .

Graphing f ʹ from f3. The following table gives the approximate

distance traveled by a downhill skier after t seconds 0 < t < 10. Make a scatter plot of the data and use that to sketch a graph of the derivative, and then answer the following .

Skiing Distances

Time t (sec)

Distance traveled (feet)

0 0

1 3.3

2 13.3

3 29.9

4 53.2

5 83.2

6 119.8

7 163.0

8 212.9

9 269.5

10 332.7

a. What does the derivative represent?Speed - rate of change

b. In what units would the derivative be measured? feet/sec

c. Can you guess an equation of the derivative by considering the graph?

015.03.3 2 xxf xxf 6.6

One-sided Derivatives

0

0

A function is if

it has a derivative at every interior point on the interval, and if the limits

lim the

lim

h

h

y f x

f a h f a

hf b h f b

differentiable on a closed interval a, b

right - hand derivative at a

the

exist at the endpoints. In the right-hand derivative, is positive and

approaches a from the right. In the left-hand derivative, h is negative and

approaches

hh a h

b h

left - hand derivative at b

from the left.b

Right-hand and left-hand derivatives may be defined at any point of a function’s domain.

The usual relationship between one-sided and two-sided limits holds for derivatives. Theorem 3, Section 2.1, allows us to conclude that a function has a (two-sided) derivative at a point if and only if the function’s right-hand and left-hand derivatives are defined and equal at that point.

Example One-sided Derivatives5. Show that the following has left-hand and right-hand derivatives at x = 0, but no derivative there.

0 ,12

10 ,1 2

xx

xxy

h

fhfh

00lim

0

h

hh

11lim

2

0

hh

0

lim 0

h

fhfh

00lim

0

h

h

h

1121

lim0

2

1lim

0h 2

1 The derivatives are not equal at x = 0.

Example One-sided Derivatives6. Show that the following has left-hand and right-hand derivatives at x = 10, but no derivative there.

1 ,241 ,2

xxxxxy

h

fhfh

11lim

0

h

hhh

21lim

2

0

3lim0

h

h3

h

hh

44lim

0

4lim0

h

4

The derivatives are not equal at x = 1.

h1 11

2 24

Pg. 92, 2.4 #2-40 even