AP Calc AB Chapter 3 Terms

f(x) is increasing when

f'>0

f(x) is decreasing when

f'<0

f(x) has a critical point when

f'=0 or f' is undefined

f(x) is concave up when f' is ____ or f’’ is ____

increasing, f’’>0

f(x) is concave down when f' is ____ or f’’ is ___

decreasing, f’’<0

f(x) has an inflection point at (c, f(c)) when

the concavity changes at f’’(c)=0 or f’’(c) is undefined

By the 1st derivative test, a point is at a relative minimum when

f' changes from negative to positive

By the 1st derivative test, a point is at a relative maximum when

f' changes from positive to negative

By the 2nd derivative test, a point is at a relative maximum at x=c if

f'(c)=0 and f’’<0

By the 2nd derivative test, a point is at a relative minimum at x=c if

f'(c)=0 and f’’>0

Theorem: If f has a relative maximum or a relative minimum at x=c, then c is

a critical number of f

In order to check for an ABSOLUTE extrema on a CLOSED interval you must check

the critical points AND the endpoints of the interval

Rolle's Theorem states that

If f(a)=f(b), f(x) is differentiable on (a,b), and continuous on [a,b], then f'(c)=0 for c∈(a,b)

The Mean Value Theorem states that

If f(x) is differentiable on (a,b), and continuous on [a,b], then f'(c)=[f(b)-f(a)]/(b-a) for c∈(a,b)

Vertical asymptotes occur when

denominator = 0 but numerator ≠ 0

Horizontal Asymptotes occur when

lim[x→±∞] f(x) = L. The asymptote is y=L.

In a rational expression, when the power of the denominator exceeds the power of the numerator, there is a horizontal asymptote at _

y=0

In a rational expression, when the power of the denominator equals the power of the numerator, there is a horizontal asymptote at _

the ratio of the leading coefficients of the numerator and the denominator

In a rational expression, when the power of the numerator exceeds the power of the denominator by one, there is a _

slant asymptote found by dividing the expression

lim[x→0] (sinx/x) =

0

Graphically a parabola has a maximum or minimum at its vertex where x=__

x=-b/2a

In order to approximate using differentials, f(x+∆x)≈

f(x)+f'(x)∆x

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval

Process of solving an optimization problem

1. Draw a sketch

2. Write an equation for the item that you want to optimize

3. Write 2nd equation to eliminate a variable by SUBSTITUTION

4. Find the DOMAIN of the function

5. Find the MAXIMUM or MINIMUM

a. Use 1st derivative test for open interval

b. Use table for closed interval