calc final

How to find vertical asymptotes in rational functions

Any x-value that…

• makes the denominator equal zero

• does not cancel out in the denominator in the rational function

How to find horizontal asymptotes in rational functions

Compare degrees of numerator and denominator

What is the HA if degrees of denominator and numerator are the same

Divide leading coefficients (num/den)

What is the HA if the numerator’s degree is greater than the denominator

No HA

What is the HA if the denominator’s degree is greater than the numerator’s

HA: y=0

What is the first step to curve sketching

Find the first derivative

What is the second step to curve sketching

Set the numerator to f’(x)=0, denominator to f’(x)=und

What do you do after finding x-values

Set the intervals and test x-values within the intervals with the first derivatives
Label whether the graph is dec/inc. during those intervals

What is the third step to curve sketching

Find the second derivative

What do you do after finding the second derivative

Find f’(x)=0 (num) and f’(x)=und. (den)

How do you test second derivative

Set intervals, test x values within intervals, evaluate CCU/CCD

How to find HAs/end behavior when curve sketching

Find limit as x approaches +- inf.
(RATIONAL) Divide by highest power of x in the denominator

(FUNCTIONS W/O DEN) Directly substitute infinities

Can radical/rational functions have more than two HA

Yes

What does POI mean

Point of inflection - concavity changes

Reasons that limits do not exist

End behaviors differ L/R
Inc./dec. w/o bound
Oscillates between 2 fixed y values

Definition of continuity part 1

f[c] must be defined

Definition of continuity part 2

lim x→c f[c] must exist

Definition of continuity part 3

f[c] = lim x→c f(x)

Types of discontinuities

Removable discontinuity: hole

Nonremovable discont: jump/VA