1)Find the Domain of the Following
MTH133
Unit 5 Individual Project – A
Name:
1)Find the domain of the following:
a)
Answer: all real numbers
Explain how you obtained your answer here:
There are no numbers that can make this function undefined.
b)
Answer: x > -3
Show your work or explain how you obtained your answer here:
log is only defined for positive numbers, and this is shifted to the left by 3 units
c)
Answer: all real numbers
Explain how you obtained your answer here:
There are no numbers that can make this function undefined.
d)
Answer: t > 1
Show your work or explain how you obtained your answer here:
ln is only defined for positive numbers, and this is shifted to the right by 1 unit
2)Describe the transformations on the following graph of. State the placement of the horizontal asymptote and y-intercept after the transformation. For example, horizontal shift to theleft 1 or reflected about the y-axis are descriptions.
a)
Description of transformation: upwards shift of two units
Equation(s) for the Horizontal Asymptote(s): y = 2
y-intercept in (x,y) form: (0, 3)
because e^0 +2 = 1+2 = 3
b)
Description of transformation: reflection about the x axis
Equation(s) for the Horizontal Asymptote(s): y = 0
y-intercept in (x,y) form: (0, -1)
because –e^0 = -1
3)Describe the transformations on the following graph of. State the placement of the vertical asymptote and x-intercept after the transformation. For example, vertical shift up 2 or reflected about the x-axisare descriptions.
a)
Description of transformation: a shift to the right by 3 units
Equation(s) for the Vertical Asymptote(s): x = 3
x-intercept in (x,y) form: (4, 0)
because log(4-3) = log(1) = 0
b)
Description of transformation: reflection about the y axis
Equation(s) for the Vertical Asymptote(s): x = 0
x-intercept in (x,y) form: (-1, 0)
because log(-(-1)) = log(1) = 0
4)The formula for calculating the amount of money returned for an initial deposit into a bank account or CD (certificate of deposit) is given by
A is the amount of the return.
P is the principal amount initially deposited.
r is the annual interest rate (expressed as a decimal).
n is the number of compound periods in one year.
t is the number of years.
Carry all calculations to six decimals on each intermediate step, then round the final answer to the nearest cent.
Suppose you deposit $2,000 for 5 years at a rate of 8%.
a) Calculate the return (A) if the bank compounds annually (n = 1). Round your answer to the hundredth's place.
Answer: $2938.66
Show work in this space. Use ^ to indicate the power or use the Equation Editor in MS Word.
A = P*(1+r/n)^(nt)
A = 2000*(1+0.08/1)^(1*5)
A = 2000*(1.08)^5
A = 2938.66
b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.
Answer: $2971.89
Show work in this space:
A = P*(1+r/n)^(nt)
A = 2000*(1+0.08/4)^(4*5)
A = 2000*(1.02)^20
A = 2971.89
c)Does compounding annually or quarterly yield more interest? Explain why.
Answer: quarterly
Explain: Quarterly compounding will give you more interest, since it gives you extra interest on the interest each quarter. That gives a higher total return.
d)If a bank compounds continuously, then the formula used is
where e is a constant and equals approximately 2.7183.
Calculate A with continuous compounding. Round your answer to the hundredth's place.
Answer:$2983.65
Show work in this space:
A = 2000*e^(0.08*5)
A = 2000*e^(0.4)
A = 2983.65
e) A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place.
Answer:8.66 years
Show work in this space:
Set A = 2P
2P = Pe^(rt)
2 = e^(0.08t)
Ln of both sides:
Ln(2) = 0.08t
Divide by 0.08:
T = ln(2)/0.08
T = 8.66
5)Suppose that the function represents the percentage of inbound e-mail in the U.S. that is considered spam, where x is the number of years after 2000.
Carry all calculations to six decimals on each intermediate step when necessary.
a)Use this model to determine the percentage of spam in the year 2003. Round your answer to two decimals places.
Answer:P = 62.44%
Show your work in this space:
plug in x = 3:
P = 13 + 45*ln(3)
P = 13 + 49.437552
P = 62.437552
Rounds to:
P = 62.44%
b)Use this model to determine in how many years (to two decimal places) it will take for the percent of spam to reach 95% provided that law enforcement regarding spammers does not change.
Answer:6.19 years
Show your work in this space:
Set P = 95:
95 = 13 + 45ln(x)
Subtract 13:
82 = 45ln(x)
Divide by 45:
ln(x) = 1.822222
Take e to each side:
x = e^1.822222
x = 6.185587
Rounds to:
x = 6.19 years