Math 315 Section 1 Fall 2006 Midterm Exam 2 Solutions 1. Give a simple example of each of the following, or argue that such a request is impossible. (a) A Cauchy sequence with a divergent subsequence. This is impossible. Every subsequence of a convergent
Proof of Various Integral
Facts/Formulas/Properties
In this section weve got the proof of several of the properties we saw in the Integrals Chapter as
well as a couple from the Applications of Integrals Chapter.
Proof of :
where k is any number.
This is a
Volumes of Solids of Revolution / Method of
Cylinders
In the previous section we started looking at finding volumes of
solids of revolution. In that section we took cross sections that
were rings or disks, found the cross-sectional area and then used
the
Annuities
Your Name: _
Topic 4. Annuities and Amortization
Many contracts use a series of small, equal payments, at equal time intervals,
to repay a large sum. For example, a series of fixed payments, monthly for 30
years, can pay off the $300,000 mortgag
Simple Discount
Your Name: _
The Simple Discount Formula is: DiscAmt = M d T
where:
the ending (future) value is M, such as $500 .
the discount rate is d, such as 2% or 0.02 (per year)
the length of time is T, such as 3 years
the starting value is M minus
Time Series
Prepared by: _
Growth Factor = NewValue / OldValue
Stock market
DIA weekly price
Jan-Apr 2015
Jan 2
Jan 9
Jan 16
Jan 23
Jan 30
Feb 6
Feb 13
Feb 20
Feb 27
Mar 6
Mar 13
Mar 20
Mar 27
$ / share
value
178
177
174
growth factor
this / previous
-0.9
Arithmetric Series
An Arithmetic Series has "n" terms. It starts with "a",
adds a constant difference "d" for each term, and ends with "z".
The first term is
a.
The second term is a + d .
The third term is
a + d(2) .
The k-th term is
a + d(k-1)
for any k=
Simple Interest
Your Name: _
The Simple Interest Formula is: FV = P ( 1 + R T )
where:
the starting value is P (Principal) or PV (Present Value), such as $100 .
the interest rate is R, such as 1.4% or 0.014 (per year)
the length of time is T, such as 3 ye
Spreadsheets
Your Name: _
For an annuity, the Future Value Factor is: s(n,i) = [(1+i)n -1] / i
where i is the effective interest rate and
the annuity is n payments of pmt dollars (at end of month or year).
The Present Value Factor is: a(n,i) = s(n,i) / (1
Geometric Series
[There was a typographical error in HW9:
Adding a constant "d" to find the next term is called an Arithmetic Series]
A Geometric Series has "n" terms, starting with "a".
Multiply by a constant ratio "r" to find the next term.
The first te
Income Statement Prepared by: _
Profit = Income - Expense
Sam Client
Income Statement
Year ending Dec 31
actual
2014
estimated
2015
Income
Salary
Interest Income
Dividends
Capital Gain
Other Income
Comments
43,000
?
?
subtotal
Expense
Income Tax (20%)
She
Time Series
Prepared by: _
Growth Factor = NewValue / OldValue
Stock market
DIA weekly price
Jan-Apr 2015
Jan 2
Jan 9
Jan 16
Jan 23
Jan 30
Feb 6
Feb 13
Feb 20
Feb 27
Mar 6
Mar 13
Mar 20
Mar 27
$ / share
value
178
177
174
growth factor
this / previous
-0.9
Net Worth Statement
Prepared by: _
Net Worth = Assets - Liabilities
Sam Client
Net Worth Statement
As of Dec 31
Assets
Checking Account
Savings Account
Bonds
Stocks
Real Estate
Car
Other Assets
subtotal
Liabilities
Home Mortgage
Auto Loan
Student Loan
Oth
More Volume Problems
In this section were going to take a look at some more volume
problems. However, the problems well be looking at here will
not be solids of revolution as we looked at in the previous two
sections. There are many solids out there that