BT1093 Matematik Perniagaan (Business Mathematics)

1
BT1093 Matematik Perniagaan(Business
Mathematics)

• Week 12, Semester 1, 2006/2007
• School of Business Economics, Universiti
  Malaysia Sabah

2
Integration by Parts Formula
If we let u f (x) and v g (x) in the
preceding formula the equation transforms into a
more convenient form
3
Integration by Parts Formula
This method is useful when the integral on the
left side is difficult and changing it into the
integral on the right side makes it easier.
Remember, we can easily check our results by
differentiating our answers to get the original
integrals.
Lets look at an example.
4
Example
Our previous method of substitution does not
work. Examining the left side of the integration
by parts formula yields 2 possibilities.
Lets try option 1.
5
Example - continued
We have decided to let u x and dv e x dx
(Note du dx and v e x)
Which is easy to integrate
x e x e x C
As mentioned this is easy to check by
differentiating.
6
Selecting u and dv
1. The product u dv must equal the original
integrand.
2. It must be possible to integrate dv by one of
our known methods.
7
Example 2
Let u ln x and dv x 3 dx, yielding
du 1/x dx and v x 4/4, and the
integral is
Which when integrated is
Check by differentiating.
8
Example 3
Let u ln y and dv dy, yielding
du 1/y dy and v y
Check by differentiating.
9
Using Definite Integrals for Average Values
Average Value of a Continuous Function f over a,
b.
10
Example
The total cost (in dollars) of printing x
dictionaries is C (x) 20,000 10x
a. Find the average cost per unit if 1000
dictionaries are produced
Note that the average cost is
30
Continued on next slide.
11
Example - continued
The total cost (in dollars) of printing x
dictionaries is C (x) 20,000 10x
b. Find the average value of the cost function
over the interval 0, 1000
20,000 5,000 25,000
Continued on next slide.
12
Example - continued
The total cost (in dollars) of printing x
dictionaries is C (x) 20,000 10x
c. Write a description of the difference between
part a and part b
From part a the average cost is 30.00
From part b the average value of the cost is
25,000.
The average cost per dictionary of making 1,000
dictionaries us 30. The average total cost of
making between 0 and 1,000 dictionaries is
25,000.
13
Continuous Income Stream
Total Income for a Continuous Income Stream.
If f (t) is the rate of flow of a continuous
income stream, the total income produced during
the time period from t a to t b is
14
Example
Find the total income produced by a continuous
income stream in the first 2 years if the rate of
flow is f (t) 600 e 0.06 t
10,000 (e 0.12 1)
? 1,275
15
Future Value of a Continuous Income Stream
From the previous example and previous work we
are familiar with the continuous compound
interest formula A Pert.
Future Value of a Continuous Income Stream
If f (t) is the rate of flow of a continuous
income stream, 0 ? t ? T, and if the income is
continuously invested at a rate r compounded
continuously, the the future value, FV, at the
end of T years is given by
16
Example
Lets continue the previous example where f
(t) 600 e 0.06 t Find the Future Value in 2
years at a rate of 10.
r 0.10, T 2, f (t) 600 e 0.06t
1,408.59
(600) (1.22140) (1.92209)
17
Other methods of integration
Integration by partial fractions Integration by
tables Differentiation equations
18
Tutorial Questions
15.1 ? 21,33,34 15.3 ? 59,61,62 15.4 ?
9,10,11,13