MATH 112
REVIEW FOR EXAM II
OPTIMIZATION
I. Derivative Rules
•There will be a page of derivatives on the exam. Know how to apply all the
derivative rules. (WS 12 and 13)
II. Functions of One Variable
•Be able to distinguish between local and global optima.
•Be able to find the global maximum and minimum of a function y=f(x) on
the interval from x=ato x=b, using the fact that optima may only occur
where f(x) has a horizontal tangent line and at the endpoints of the interval.
Step 1: Compute f0(x).
Step 2: Find all values of xat which f0(x) = 0.
Step 3: Plug all the values of xfrom Step 2 that are in the interval from ato
band the endpoints of the interval into the function f(x).
Step 4: Sketch a rough graph of f(x) and pick off the global max and min.
•Be familiar with the following two applications:
–Maximizing T R(q) starting with a demand curve. (WS 15)
–Optimizing the slope of a diagonal line through a given curve. (WS 16)
•Understand how to use the Second Derivative Test. (WS 17)
If f0(a) = 0 and:
–f00(a)>0, then f(x) has a local min at x=a.
–f00(a)<0, then f(x) has a local max at x=a.
–f00(a) = 0, then the test tells you nothing.
IMPORTANT!!! For the Second Derivative Test to work, you must have
f0(a) = 0. If f00 (a)>0 but f0(a)6= 0, then the graph of f(x) is concave up at
x=abut f(x) does not have a local min there.
III. Functions of Two Variables
•Be able to compute overall, incremental, and instantaneous rates of change of a
function of two variables. (WS 18A)
•Be able to compute partial derivatives using all the derivative rules.
•Know how to find the candidates for maxima and minima in a function of two
variables. (Take both partial derivatives, set them equal to 0, and solve the
resulting system of equations.)
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