The Graph of the Derivative

I.  Quiz

II.  Homework

III.  Liebnitz notation and the right triangle

We will see how the slope of the secant line can be interpreted as (Delta y)/(Delta x)

taking a limit as Delta x approaches 0 we arrive at the derivative and call it

f'(x) = dy/dx.

IV.   Constructing the graph of the derivative from the graph of a function

We will practice sketching the graph of the derivative from the graph of the function.  Several examples will be given in class.  We will also go backwards, that is sketch the graph of a function knowing the graph of the derivative.  

Exercise:  A trip from San Francisco to South Tahoe.

Suppose that you are driving from SF to Tahoe.  You begin your trip on the 101 travelling 50 mph, you hit stop and go traffic at the bay bridge.  Through Oakland and Berkeley, there is heavy traffic.  From Berkeley to Vallejo there is no traffic, but the speed limit is 55, so you go 65.  From Vallejo to Vacaville the limit is 65, so you set your cruise control to 70.  At Vacaville you stop at the In and Out for a Burger and munch of 30 minutes.  Then its smooth sailing at 70 until Placerville where you get stopped in traffic.  You continue up the mountain slowing down whenever the road becomes curvy, until you arrive at South Lake Tahoe.  Let s(t)be the function that relates time to distance travelled.

A)  Sketch the graph of f'(t)

B)  Sketch the graph of f(t).

V.  Sums and Products of Differentiable Functions

Theorem:  If f and g are differentiable then

A)   [f + g]' = f' + g'

B)  [f - g]' = f' - g'

C)  [cf]' = c(f')

Proof of C)

lim [(cf(x + h) - cf(x))/h] =

lim [c(f(x + h) - f(x))/h] =

c lim [(f(x + h) - f(x))/h] =

cf'(x).

Example:  

We know that if f(x) = x² then f'(x) = 2x and that if f(x) = x then f'(x) = 1 so that if

h(x) = 4x² -3x

then

h'(x) = 4(2x) - 3(1) = 8x - 1