Test 3 Review
Sections 3.11, 4.1 - 4.3, 4.5, 4.7
Critical points
Where f โ(x)=0 or f โ(x) is undefined (must be in the domain of f (x)).
Absolute extrema
A continuous function on a closed interval always has an absolute max and an absolute min.
Absolute extrema may occur at critical points or endpoints.
Test critical points and endpoints in function.
Local extrema
Occur at critical points
First derivative test:
Where f โ(x) changes from + to โ f(x) has a local max
Where f โ(x) changes from โ to + f(x) has a local min
Second derivative test:
If f โ(a) = 0 and
f โ(a) > 0, f(a) is a local min
f โ(a) < 0, f(a) is a local max
Monotonicity
f(x) increasing when f โ(x) > 0
f(x) decreasing when f โ(x) < 0
Concavity
f(x) concave up when f โ(x) > 0
f(x) concave down when f โ(x) < 0
f(x) has a point of infection at x = a if
f(x) changes concavity at x = a and x = a is in the domain of f(x)
Asymptotes
Vertical Asymptotes
x= a is a VA of f(x) if ยฑโ=
ยฑ
โ
)(lim xf
ax
Horizontal Asymptotes
y = b is a HA of f(x) if bxf
x
=
ยฑโโ )(lim
Scheme for Graphing
a) Determine the Domain.
b) Determine the asymptotes (vertical, horizontal).
c) Determine the x- and y- intercepts.
d) Determine the critical point(s) (Set fโ=0 and undefined).
e) Determine the intervals where the function f is increasing/decreasing.
f) Determine the local extrema.
g) Determine the possible point(s) of inflection (Set fโ=0 and undefined).
h) Determine the intervals where the function f is concave up/down.
i) Determine the inflection point(s).
j) Sketch the graph using the information obtained above.
