Warmup

Find k such that the line is tangent to the graph of the function

9-4x y :line )( 2 kxxxf

-10k -3,x when , 2 k 3, x

3

9x

94)42(x

4-2xk

42

2

2

when

x

xxx

so

kx

Warmup:

x

xx

24lim

0

3.1 Derivatives

Nope, not thatkind ofderivative

Stock Market/Economic Crash

0

limh

f a h f a

h

is called the derivative of at .f a

We write: 0

limh

f x h f xf x

h

“The derivative of f with respect to x is …”

Alternate Form of Derivative

lim ( ) ( )'( ) x c

f x f cf c

x c

provided the limit exists

There are many ways to write the derivative of y f x

f x “f prime x” or “the derivative of f with respect to x”

y “y prime”

dy

dx“the derivative of y with respect to x”

df

dx“the derivative of f with respect to x”

df x

dx“the derivative of f of x”( of of )d dx f x

2'(1) ( ) 2Find f for f x x

dx does not mean d times x !

dy does not mean d times y !

dy

dx does not mean !dy dx

df x

dxdoes not mean times !

d

dx f 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

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.

Graphing a Derivative on TI 83+

                                                              

The proper notation for graphingthe derivative is nDeriv(function,X,X).

Graph the derivative of the function              

Enter the functionand derivative.

Use TABLE to help find a window.

Set a window.

Use the Thick option for the derivative andgraph the functions.

Graph of f(x) Make a table of approximations of slopes of tangent lines at the pts

Point x slope

A 0

B 1.5

C 2.5

D 3

E 5

F 6

? 7

6

1

0

-1.5

-1

0

1

Now lets take these values and make a graph of all the slopes ( f ‘ (x) graph )

Connect them for your derivative graph

A’

B’

C’

D’ E’

F’

?’

xy Sketch y

Conceptual questions:

Let y = g(x) be a function that measures the water depth in a pool x minutes after the pool begins to fill. Then g’(25) represents:

I. The rate at which the depth is increasing 25 minutesafter the pool starts to fill

II. The average rate at which the depth changes overthe first 25 minutes

III. The slope of the graph of g at the point where x = 25

A) I only B) II only C) III only D) I and II

E) I and III F) I, II, and III

The function y = f(x) measures the fish population in Lincoln Pond at time x, where x is measured in years sinceJanuary 1st, 1950. If : thatmeansit ,500)25( f

A) There are 500 fish in the pond in 1975

B) There are 500 more fish in 1975 than there were in 1950

C) On average, the fish population increased by 500 per year over the first 25 years following 1950

D) On Jan. 1st, 1975, the fishing population was growingat a rate of 500 fish per year

E) None of the above

f(x) = position function

f’(x) = velocity function

f”(x) = acceleration function

The end

• p. 101 (1-10, 12, 16, 18, 25, 26 a-e, 28)