Clac exam 2

How to find Absolute Max & Min

1) take 1st derivative

2) set it = 0 & solve for x (crit #)

3) Chart end point & crit #’s

4) plug those into original function

-Format = Abs. Max/Min ___ at x/y = ___

Any positive base

raised to a negative power is always positive

-e⁰ =

-ln(1) =

-e⁰ = 1

-ln(1) = 0

What cant be in the domain

1) no divided by 0

2) no negative in √

3) ln(x) cannot be negative or 0

Log rules

arctan

arcsin

Whats the derivative of

1) ln(x)

2) ln(stuff)

3) e^stuff

Whats the derivative of

1) θ

2) π

-1

-0

How to get rid of “e” or “ln” when solving for x

Add ln or e to get rid of the other

exponent rules

Log differentiation rules

1) write y = f(x)

2) add ln(_) to both sides

3) Use log rules to simplify as much as possible

4) Take the derivative of both sides using implicit differation

5) Multiply y to both sides

6) then replace “y” with f(x)

How do you plot # line if

There’s no crit #s and excluded domain #s

Put the domain on each end of the line and test any # in between for intervals of concavity or increasing/decreasing

if you set a equation to zero and get

a # = 0

then there’s no crit #’s

if there’s no crit #’s when looking for local max or min

then there’s no local Max or min

How to find intervals of increasing and decreasing

1) find 1st derivative crit #’s

2) make number line plotting crit #’s & any excluded domain #’s

3) Test each side using 1st derivative function

How to find intervals of concave up & down

How to find inflection point

1) find 2nd derivative crit #’s

2) make number line plotting crit #’s & any excluded domain #’s

3) test each side using the 2nd derivative function

Inflection point ( x,y )

1) anywhere the sign changes from “-” to “+” or vice versa

2) plug that x into original function to find y-coordinate

3) check that it fits in the domain

if the y-coordinate of a inflection point is undefined

it is not a inflection point

How to find y-coordinate of local max/min

plug the x into the original function and solve

x² = a negative number

Never true (so no x value)

Chain rule formula

When using implicit differentiation add

* y’ anytime you take the derivative of y

Local max and min

Only occur at interior points of a interval

What is a critical number or point

Does a sharp corner still make a extreme

Where f ‘(c) = 0 or DNE

Yes it does

On a graph showing f ‘ (x)

-How to find f local max

-How to find f local min

-How to find f increasing and decreasing

-Where f ‘ (x) changes from positive to negative (on x axis)

-Where f ‘ (x) changes from negative to positive (on x axis)

-Where f ‘ (x) is above or below x axis

On a graph of f ‘ (x)

1)How to determine intervals of concavity for f

2)how to determine inflection points for f

1)f is concave up when f ‘ (x) is increasing

• f is concave down when f ‘ (x) is decreasing

2) I.P. where f ‘ (x) changes from increasing to decreasing or vice versa

How to find local max, min, or neither

1) find 1st derivatives crit #’s

2) make a number line plotting crit #’s & any excluded domain #’s

3) test each side using 1st derivative function to see if its positive or negative

4) then draw increasing or decreasing arrow for each test to see if the points are max, mins, or neither

concave up

concave down