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Aim: What is the relationship between the graph of a derivative function and the graph of the original function?. Do Now:. Sketch the derivative of the function given by the following graph:. f ( x ). f’ ( x ). Sketching the Derivative. derivative is positive when original is increasing.

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

Do Now:

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


Sketching the derivative l.jpg

f derivative function and the graph of the original function?(x)

f’(x)

Sketching the Derivative

derivative is positive when original is increasing.

derivative is negative when original is decreasing.

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

pay attention to steep parts.


Derivative graphs l.jpg
Derivative Graphs derivative function and the graph of the original function?

Sketch the graph of this function’s derivative


Derivative graphs4 l.jpg
Derivative Graphs derivative function and the graph of the original function?

Sketch the graph of this function’s derivative


Derivative graphs of position equation l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

Sketch the graph of this function’s derivative


Derivative graphs6 l.jpg
Derivative Graphs derivative function and the graph of the original function?

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


Derivative graphs of position equation7 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

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

B = position, A = velocity, C = acceleration

Which graph is which?


Derivative graphs of position equation8 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

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

A = position, C = velocity, B = acceleration

Which graph is which?


Derivative graphs of position equation9 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

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

C = position, B = velocity, A = acceleration

Which graph is which?


Derivative graphs of position equation10 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

Graph shows velocity v as function of time t.

2 < t < 3, 6 < t < 7

When is the body's acceleration equal to zero?


Derivative graphs of position equation11 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

Graph shows distance s as function of time t.

1 < t < 2

When is the body standing still?


Derivative graphs of position equation12 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

y = h’(x)

On what interval(s) is the function h positive?

  • -2 < x < -.6

  • (-2, -.6)  (1, 1.6)

  • (-, -2)

  • All x values

  • Impossible to tell


Derivative graphs of position equation13 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

y = h(x)

What are the zeros of g’(x)?

  • x = -2, -0.6

  • x = -2, -0.6, 1, 1.6

  • x = -1.6, 1.3

  • x = -1.6, 0.1, 1.3

  • Impossible to tell


Derivative graphs of position equation14 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

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

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

  • h’ has no zeros

  • h’ is undefined at x = 1

  • h’ is always negative on its domain

  • h’ has no smallest possible value

  • all of the above


Derivative graphs of position equation15 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

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

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

  • h’ has no zeros

  • h’ is undefined at x = 1

  • h’ is always negative on its domain

  • h’ has no smallest possible value

  • all of the above


Derivative graphs of position equation16 l.jpg
Derivative Graphs of Position Equation derivative function and the graph of the original function?

no relative max or min – no zero slope

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

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


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