## Lecture Notes: Wednesday, October 21, 1998

Sixth Homework
Assignment - for discussion at recitations Tuesday October 20, and
Thursday October 22. The sixth homework also includes your first
**MAPLE ASSIGNMENT**.

### Chapter 4 (Sections 6 and 7)

We continued our work on optimization.

Section 4.7. Problems 22, 24.

MAPLE Demonstration
#5 This is a particularly good demo. Please be sure to work
through it.

Additional examples:

1. The Manager of a 100 unit apartment complex knows from
experience that all units will be occupied if the rent is $400 per
month, and, on average, one additional unit will remain vacant for
every $5 increase in rent. What rent should be charged to maximize
revenue? (We found the answer to be $450.)

2. A bookstore is attempting to determine the economic order
quantity for a popular book. The store sells 8000 copies of the book
each year. The store figures it costs $40 to process each order for
new books. The carrying cost (due primarily to interest payments) is
$2 per book per year, to be figured on the maximum inventory during
an order-reorder period. How many times a year should orders be
placed? (We found the answer to be 20 orders per year.) **Special
note**: In this problem the carrying cost depends on the *maximum
inventory*. In the problems 22 and 24 above, the carrying or
storage costs depend on the *average inventory* *size*.

3. **Challenge problem**: What is the largest volume of a right
circular cylinder inscribed in a sphere of radius r? The answer will
depend on r. Can you determine that the answer is
(1/sqrt(3))*((4/3)*Pi*r^3)? Note that this is the volume of the
sphere divided by the square root of 3.

### Chapter 5

**Exponential Functions**

b positive real number, not 1. (Think of b = 2 case.)

2^n = 2*...*2 n times (n an integer)

2^(1/m) = a means a^m = 2 (m an integer)

2^(n/m) makes sense.

Since we can approximate arbitrary real numbers arbitrarily
closely by rationals, we can make sense of 2^x for any real number
x.

Properties:

(b^x)*(b^y) = b^(x+y)
(b^x)^y = b^(x*y)

(a*b)^x = (a^x)*(b^x)

b^x is called the exponential function with base b.

In MAPLE, we plotted the cases b = 1/2, 1, 2. Note domain, range,
slope, y-intercept.

Definition: e = lim(as n -> infinity) (1+ 1/n)^n We saw in
MAPLE that this limit gives e and that the approximate value of the
irrational number e is 2.71828....

f(x) = e^x = exp(x) is called THE exponential function. It turns
out this is the only exponential function (i.e. choice of base) whose
slope at x= 0 is equal to 1.

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