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Review, Part 2
L’Hopital’s Rule
x
e 1
Ex. lim
x0 sin x
ex
Ex. lim 3
x  x
Critical Points
 Where f = 0 or is undefined
Ex. If f  x   x  x , find and classify all critical
points
4
8
3
3
 Inflection points are where the concavity of f
changes
Ex. If f  x   x  x , find all inflection points.
4
8
3
3
Absolute max/min
 Check all critical points and endpoints
Ex. If the velocity of a particle is given by
v(t) = t 3 – 3t 2 + 12t + 4, find its maximum
acceleration on the interval 0 ≤ t ≤ 3.
Strategy for Optimization
1) Draw a picture, if appropriate
2) Write down given information, including an
equation
Find the largest, smallest, closest
3) Find the function to be optimized
Maximize,
minimize
4) Substitute
to get one variable
5) Take the derivative
6) Set equal to zero and solve
Strategy for Related Rates
“PGWEDA”
P) Draw a picture
G) Identify given information, including rates
(derivatives)
“Changes at a rate of …”
W) Identify what you want to find
Involves
time
E) Find an equation
to relate
the variables
D) Take the derivative with respect to time
A) Plug in values to get your answer
Ex. The figure represents an observer at point A
watching balloon B as it rises from point C. The
balloon rises at a constant rate of 3 m/sec and the
observer is 100 m from point C. Find the rate of
change of θ when y = 50.
x
A

100
B
y
C