Math 1210-001
Tuesday Mar 1
WEB L110
Finish 3.2-3.3: monotonicity, concavity, local extrema, first and second derivative tests.
, Discuss the first and second derivative tests, for deciding whether stationary points are the locations of
local maxima, local minima, or neither. Notes for this were originally on the last page of Friday's notes,
and are reproduced here:
First derivative test: Let f x be continous in an open interval I = a, b containing c, and differentiable in
that interval except maybe at c. Then
(i) If f# x ! 0 for a ! x ! c and f# x O 0 for c ! x ! b then f c is a local minimum.
(ii) If f# x O 0 for a ! x ! c and f# x ! 0 for c ! x ! b then f c is a local maximum.
Pictorially,
Second derivative test: Let f x be differentiable on an open interval I containing c and let c be a stationary
point (f# c = 0 . If f## is continuous on I and
(i) f## c O 0 then f c is a local minimum.
(ii) f## c ! 0 then f c is a local maximum.
Pictorially,
Exercise 1) Yesterday we carefully sketched the graph of f x = x4 K 2$x3 . There was only one local
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(and global) extreme value, f
=K . Check that the first and second derivative tests apply there
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16
(and that the tests don't apply at the stationary point x = 0.
, Discuss Snell's law, as an example of the first derivative test. (This was page two of Monday's notes.)
x2 K 3 x C 2
Exercise 2) (See Webwork problem 19) Sketch the graph of f x =
. Find all asymptotes
xK3
to the graph. Find the intervals where f is inc/dec, CU/CD. Hints: First do long division to simplify the
formula for f. You can use the critical points and concavity to deduce the intervals where f is increasing
and decreasing. (This graph is complicated enough that it's really more section 3.5 material than section
3.3 material. But it's good practice.)
6
4
y
2
K2
0
K2
K4
K6
2
4
x
6
8