MBAD 51415142

1
MBAD 5141/5142

• Class 4
• Oh Yeah! More Calculus!

2
Outline for todays class

• Homework help?
• Derivative of exponential and logarithmic
  functions
• Maxima and Minima
• Application Problems

3
Derivatives of exponential functions

• An exponential function is one where the base is
  constant and the exponent is a variable. The most
  common exponential function used in business
  applications is one involving

4
Rules for derivatives of Exponential Functions
If
Then
Then
If
Then
If
5
For example
Find the derivative of
and
6
Rules for Derivatives of Logarithmic Functions
Then
If
Then
If
7
For Example
Find the derivative of
and
8
Maxima and Minima

• Sometimes it is beneficial to look at specific
  points on a function. For example, business
  owners often want to know what is the highest
  price they can charge and still not suffer loss
  of sales. Or, what is the lowest cost that can be
  achieved while still maintaining production? To
  accomplish this we must look at turning points in
  the functions. These are called maximum and
  minimum points (plural extrema). We can use
  calculus techniques to determine these points.

9
Maxima and Minima (continued)

• It is important to stop here and review some
  function definitions
• A function is increasing if there is a
  corresponding increase in y-values for every
  increase in x-values.
• If f (x) 0 then f(x) is increasing.
• A function is decreasing if there is a
  corresponding decrease in y-values for every
  increase in x-values.
• If f (x) 0 then f(x) is decreasing.
• If f (x) 0 then the function is constant
  (flat)

10
Maxima and Minima (continued)
Increasing
Decreasing
4
3
2
1
1
2
3
11
Maxima and Minima (continued)

• Definition Maxima and Minima occur at any
  critical point c. These happen where the
  function turns from increasing to decreasing or
  vice versa. The derivative of c only occurs at
  smooth turns, not sharp points.

f(c) doesnt exist
f(c) exists
12
Maxima and Minima (continued)

• Here is a way to find maximum and minimum points.
  It is known as the First Derivative Test

Let x c be a critical point of the function
f If f (x) gt 0 for x just below c and f (x)
lt 0 for x just above c, then c is a local
maximum of f. If f (x) lt 0 for x just below c
and f (x) gt 0 for x just above c, then c is
a local minimum of f. If f (x) has the same sign
for x both below and above c then there is no
local extrema of f.
13
Find the value of x at the local maxima and
minima of the following function
14
Maxima and Minima (continued)

• It is also possible to find local maxima and
  minima by taking a second derivative and applying
  it to a test. The test is spelled out on page 232
  of your text but it basically says that if f
  (c) lt 0 then f has a local maximum at x c. If
  f (c) gt 0 then f has a local minimum at x c.
  If f (c) 0 or fails to exist then the test
  fails.

15
Applications of Maxima and Minima Example 1
A lake is being stocked with fish. The more fish
put into the lake, the more competition there
will be for the available food supply. One result
of this is that the fish will gain weight more
slowly. It is known from previous experiments
that when there are n fish per unit area of
water, the average amount that each fish gains in
weight during one season is given by w 600-30n
grams. What value of n leads to the maximum total
production of weight in the fish?
16
Applications of Maxima and Minima Example 2
A small manufacturing firm can sell all the items
it can produce at a price of 6 each. The cost of
producing x items per week is (in dollars) is
What value of x should be selected in order to
maximize profits?
17
Applications of Maxima and Minima Example 3
The cost of producing x items per week is
For the particular item in question, the price at
which x items can be sold per week is given by
the demand equation
Determine the price and volume of sales at which
the profit is maximized.