Comp book: MVT / RT / Extrema / Concavity / Templates

Question 1

Mean Value Theorem

Answer 1

Asking if the average rate of change will equal the instantaneous rate of change.

1. Make sure it is continuous and differentiable
2. Find the average rate of change
3. Find the derivative equation
4. Set them equal
5. Solve

Question 2

Rolles Theorem

Answer 2

Finds if there is a max/min

1. Check to see if it continuous and differentiable
2. Check if f(a) = f(b)
3. Find the derivative
4. Set to 0
5. Solve

Question 3

Extrema:
Absolute extrema (closed intervals)

Answer 3

1. Find f’
2. Find critical numbers
   f’ = 0 or DNE
3. Make a table
   End point critical point End point
   F(x)
4. Plug into f(x)

Question 4

Extrema:

Relative extrema

Answer 4

1. Find f’
2. Find critical points
3. Make a sign chart and plug into f’
   f’ = + ——> -
   f’ = - ——> +

If f’ is greater than zero than f is increasing
If f’ is less than zero, than f is decreasing

Question 5

Concavity:
POI
Up/Down

Answer 5

To find POI:

1. Use the 2nd derivative
2. Find the critical points
3. Make a sign chart using f”

Concave up when f” is positive
Concave down when f” is negative

Question 6

What does f” tell us?

Answer 6

F” > 0 :
f’ is increasing
f is concave up

F” < 0 :
f’ is decreasing
f is concave down

Question 7

Extrema and concavity templates:

Is f increasing or decreasing

Answer 7

F is _______ on the interval ______ because f’ is positive/negative

Question 8

Extrema and concavity templates:

Extrema

Answer 8

There is a min or max at _______ because f changes from _____ to ______

Question 9

Extrema and concavity templates:

Concavity

Answer 9

f is concave _______ on the interval _____ because f” is _________

Question 10

Extrema and concavity templates:

Points of Inflection

Answer 10

There is a POI at _______ because f” changes from _______ to _______