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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