Unit 5 - Analytical Application of Differentiation

Question 1

What is the Extreme Value Theorem?

Answer 1

If a function is continuous over interval [a,b], then f has at least one minimum value and at least one maximum value on [a,b].

Question 2

What is a critical point?

Answer 2

Point with the possibility of being an extrema

Question 3

How do you determine a critical point?

Answer 3

The derivative of the function does not exist or equals zero.

Question 4

How do you determine a relative extrema?

Answer 4

Isolate the suspected point as well as surrounding domain. If this point is higher or lower than the points adjacent to it, then it can be considered an relative extrema.

Question 5

What is the absolute maximum?

Answer 5

Point on a function that has the highest y-value

Question 6

What is the absolute minimum?

Answer 6

Point on a function that has the lowest y-value

Question 7

What is the Mean Value Theorem?

Answer 7

If a function f is continuous and differentiable over the interval [a,b], then there exists a point c within that open interval where the instantaneous ROC equals the average ROC over this interval.

Question 8

What is the equation for the Mean Value Theorem?

Answer 8

f’(c) = f(b) - f(a) / b - a

Question 9

When the slope of a function is positive, then the function is

Answer 9

increasing

Question 10

When the slope of a function is negative, then the function is

Answer 10

decreasing

Question 11

How to find intervals of a function that is increasing or decreasing

Answer 11

1- Find critical points
2- Between these points, intervals of derivative must be positive or negative
3- Use sign chart to keep track
4- Answer with justification

Question 12

How to justify a function is increasing on an interval

Answer 12

Claim that its derivative is positive

Question 13

How to justify a function is decreasing on an interval

Answer 13

Claim that its derivative is negative

Question 14

First Derivative Test

Answer 14

Used to determine where min/max may exist on a function

Question 15

Minimum exists when

Answer 15

the derivative of a function changes sign from negative to positive

Question 16

Maximum exists when

Answer 16

the derivative of a function changes sign from positive to negative

Question 17

“What is the maximum value” is asking for

Answer 17

f(c) or the y-value

Question 18

“What is the maximum “ is asking for

Answer 18

x=c or the x-value

Question 19

Endpoints

Answer 19

Also has the possibility of being a minimum or maximum on an interval

Question 20

Candidates Test

Answer 20

1 - Find derivative of function and set equal to 0 to find critical points
2 - Substitute critical points & endpoints into original function
3 - Lowest value is considered the abs. min. value
Highest value is considered the abs. max. value

Question 21

Conavity

Answer 21

State of being concave

Question 22

If f is concave up, then

Answer 22

f’ is increasing

f’’ > 0

Question 23

If f is concave down, then

Answer 23

f’ is decreasing

f’’ < 0

Question 24

How to find intervals of concavity

Answer 24

1- Find the second derivative and set it equal to 0

2- Use sign chart to find intervals of concavity

Question 25

Points of Inflection

Answer 25

There is a point of inflection of f at x=c if f(c) is defined and f'' changes sign at x=c Where graph changes concavity

Question 26

Two common mistakes

Answer 26

1 - Assuming that f'' = 0 means there is a point of inflection -> need to check for sign change 2 - Assuming that f'' is not equal to 0 means there are no points of inflections

Question 27

Relative max

Answer 27

f'(x) = 0 | f"(x) < 0

Question 28

Relative min

Answer 28

f'(x) = 0 | f"(x) > 0

Question 29

If there is only 1 critical point, and that critical point is an extremum, then

Answer 29

it is an absolute extremum

Question 30

The slope of f is the

Answer 30

y-value of f'

Question 31

Optimize

Answer 31

Make the best or most effective use of a situation or response

Question 32

Strategies for optimization

Answer 32

1 - Draw a picture and identify known and unknown quantities 2 - Write an equation that will be optimized 3 - Write your equation in terms of single variable 4 - Determine desired max or min value with calculus techniques 5 - Determine domain and endpoints of equation to verify if endpoints are max or min

Question 33

Implicit relationships follow the same rules as functions when it comes to

Answer 33

derivative and second derivatives if dy/dx = 0 or DNE, then point = critical point if d22y/dx^2 > 0, then graph = concave up if d22y/dx^2 < 0, then graph = concave down