Math 131 Take Home Quiz 5
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Name_________________________
_____________________________
The first three problems are about bell shaped curves.
1. A logistic function is
f (x) 
L
1  ae
 bx
for L > 0, a > 0. The derivative of f(x) is
bell shaped and the bell is symmetric only if a = 1.
We will consider the special case
Find
lim f ( x )
x 
f (x) 
lim f ( x )
x  
Find the derivative of f(x) and show all work.
Graph the derivative here.
2
1 e
x
. Graph this function here:
2. The standard normal density function is
f (x) 
1
2

e
1 2
x
2
. Graph f(x) here.
Find
f ( x )
and graph it.
Find
f ( x )
and locate the x-values of the inflection points. Show your work.
3.
g(x) 
3
3 x
2
Graph g(x) here.
Find
g ( x )
and graph it here.
Find
g ( x )
showing all work and find the x-values of the inflection points.
4. A rectangular glass box has metal edges. The length of the base is twice the width and
the volume of the box is 12 cubic inches.
Find the dimensions of the box that minimize the amount of metal used. Use the 2nd
derivative test to verify you found the minimum.