Aim: Functions & Derivative Graphs Course: AP Calculus Do Now: Aim: What is the relationship between the graph of a derivative function and the graph of.

Aim: Functions & Derivative Graphs Course: AP Calculus

Do Now:

Aim: What is the relationship between the graph of a derivative function and the graph of the original function?

Sketch the derivative of the function given by the following graph:


Sketching the Derivative

derivative is positive when original is increasing.

derivative is negative when original is decreasing.

pay attention to steep parts.

f(x)

derivative is zero when original is at relative maximum/minimum.

10

5

-5

-10

-5 5

f’(x)


Derivative Graphs

Sketch the graph of this function’s derivative

15

10

5

-5

-10

-15

-20 -10 10


Derivative Graphs


15

10

5

-5

-10

-15

-20 -10 10


Derivative Graphs of Position Equation


15

10

5

-5

-10

-15

-20 -10 10


6

4

2

-2

-4

-6

-5 5

Derivative Graphs

Use the following info to graph f over [-5, 6] 1) f is made of closed line segments end to end 2) graph starts at point (-5, 1) 3) derivative of f is step function shown here



Which graph is which?

Graph shows distance s, velocity v, and acceleration a as function of time.

B = position, A = velocity, C = acceleration





A = position, C = velocity, B = acceleration





C = position, B = velocity, A = acceleration



When is the body's acceleration equal to zero?

Graph shows velocity v as function of time t.

2 < t < 3, 6 < t < 7



When is the body standing still?

Graph shows distance s as function of time t.

1 < t < 2



y = h’(x)

On what interval(s) is the function h positive?a. -2 < x < -.6

b. (-2, -.6) (1, 1.6)

c. (-, -2)

d. All x values

e. Impossible to tell



y = h(x)

What are the zeros of g’(x)?a. x = -2, -0.6

b. x = -2, -0.6, 1, 1.6

c. x = -1.6, 1.3

d. x = -1.6, 0.1, 1.3

e. Impossible to tell



a. h’ has no zeros

b. h’ is undefined at x = 1

c. h’ is always negative on its domain

d. h’ has no smallest possible value

e. all of the above

This is the graph of y = h(x).

What can you expect concerning the function h’(x)?



a. h’ has no zeros

b. h’ is undefined at x = 1

c. h’ is always negative on its domain

d. h’ has no smallest possible value

e. all of the above

This is the graph of y = h(x).

What can you expect concerning the function h’(x)?



no relative max or min – no zero slope

at x > 0 slope is very (-) and gets less negative (-); slope is never positive

from l – r: slope is (-) and gets more (-) as x approaches 0 where f is undefined

Aim: Functions & Derivative Graphs Course: AP Calculus Do Now: Aim: What is the relationship between the graph of a derivative function and the graph of.

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