AP Calc AB Unit 5 part 1

What is a critical point?

a point with a possibility of being a critical point

Where are critical points?

1. where f’(x) DNE by setting denom=0

2. where f’(x)=0 by setting nun=0

How do you know if a possible critical point is in the interval?

test it in the original function

List the steps to know if a function is increasing or decreasing

1. find CPs (find deriv, set num=0 and den=0)

2. make a sign chart with CP

3. plug in numbers into the deriv that are before, between, and after CPs

4. answer and justify

How do you justify where a function is increasing/ decreasing?

“f is increasing on (#,#) b/c f’(x)>0”

OR

“f is decreasing on (#,#) b/c f’(x)<0”

On a graph of f’(x), what do you look at to see if the original function is increasing/decreasing?

the sign of its rate of change

What does the first derivative test do?

tests if a function has extrema (maximum/minimum)

How do you do the first derivative test?

1. find the derivative of the original function

2. find CPs (find deriv, set num=0 and den=0)

3. make a sign chart with CPs

4. plug in numbers into the original function that are before, between, and after CPs to see if +/-

5. draw↗arrows where positive and ↘ arrows where negative

6. identify mins and maxes and justify

How do you justify the first derivative test?

“min value at x=c b/c f’ changes sign from negative to positive

OR

“max value at x=c b/c f’ changes sign from positive to negative

Explain the difference between “What is Max value” and “Where is max?”

What is max: y-value, f(c)

Where is max: x=c

What are the candidates for absolute extrema?

critical points and endpoints

How do you find absolute extrema from candidates?

1. find the derivative

2. find CPs (find deriv, set num=0 and den=0)

3. plug in x-values to the original function to get y-values

4. the highest # = abs max and the lowest # = abs min

How do you determine the absolute extrema from the graph of f’(x)?

mark the points where f’(x) crosses the x-axis. Areas with the most shaded are the abs max (positive) and abs min (neg)

what does it imply about f’(x) and f”(x) if f(x) is concave up? What about concave down?

concave up: f’(x) is increasing and f”(x) is positive

concave down: f’(x) is decreasing and f”(x) is negative

How do you find intervals of concavity?

1. find the first derivative

2. find the second derivative

3. find points of inflection set num=0 and den=0)

4. plug in points to f”(x) to see if +/-

5. make a two level sign chart with f”(x) and f(x)

6. answer and justify

How do you justify intervals of concavity?

“concave up on (-∞, #) b/c f”>0”

AND

“concave down on (#, ∞) b/c f”<0”

What does the second derivative test do?

finds the relative extrema (min

What does it mean if there is only one min or one max?

it is the absolute min/max

How do you use the second derivative test?

1. find CPs (find deriv, set num=0 and den=0)

2. find second derivative

3. plug in CPs to second derivative to see if +/-

How do you justify the second derivative test?

“rel max at x=c b/c f’(c)=0 and f”(c)<0”

OR

“rel min at x=c b/c f’(c)=0 and f”(c)>0”