Application Of Differentiation - Ch 4

Question 1

Is f(-1) = 37 a local maximum? Is it a absolute maximum?

Answer 1

No, local cannot be at an endpoint. However this is the absolute maximum

Question 2

What are the two conditions for the extreme value theorem?

Answer 2

Continuous and on closed interval

Question 3

What are the min and max values on these graphs?

Answer 3

Question 4

Can you locate local extremes by finding where derivative equals zero?

Answer 4

No. X^3 has derivative 0 at x=0, but it’s not an extreme.

However it is true that if there is a local extreme, and the derivative exists, then derivative is zero. Formats theorem.

Question 5

Can there be a local extreme if the derivative does not exist?

Answer 5

Yes, y=|x|. Local min at 0, but derives no exists

Question 6

What’s a critical number?

Answer 6

A number c in the domain of f such that either f’(c) = 0 or does not exist

Question 7

How to find absolute max and min values of continuous function of a closed interval

Answer 7

Find f value at critical numbers
Find f value at endpoints
Largest and smallest are the extremes

Question 8

What’s the mean value theorem

Answer 8

Significance is that there is some number in the middle where the instantaneous rate of change equals the average rate of change

Question 9

How can derivatives determine if graph is concave upward or downward?

Answer 9

Question 10

Con cavity test

Answer 10

Question 11

How do you find points of inflection

Answer 11

Second derivative changes sign

Question 12

How use derivatives to find local max or min

Answer 12

Question 13

Second derivative test to determine if local min or max

Answer 13

Question 14

What’s l’hospitals rule

Answer 14

Conditions: f and g are differentiator and g’ not equal zero

The limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

Useful when limit is an indeterminate form eg 0/0 or infinity/infinity

Question 15

Answer 15

Question 16

Answer 16

Question 17

Pic

Answer 17

Question 18

Answer 18

Question 19

Are these indeterminate:

1^(infinity)

0^(infinity)

Answer 19

Yes

No (its zero)

Question 20

Are these indeterminate?

0^0
infinity^0
1^infinty

Answer 20

yes

infinity is a concept its not a number

Question 21

pic

Answer 21

Get (inifinity * 0) but LHR only applies to quotient, so need to rewrite as a fraction

Question 22

Answer 22

Question 23

Answer 23

rewrite as fraction so can use LHR.

log each side
can move log into limit bc log is continuous

Question 24

Answer 24

Question 25

Answer 25

remember its ln y… do last step

Question 26

whats rolles theroem

Answer 26

if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.