Name: Terri Woodard

MTH133

Unit 5 Individual Project – A

Name: Terri Woodard

1) Find the domain of the following:

a)

Answer: All real number

Explain how you obtained your answer here:

For , the value of t will be any real number (positive or negative). Therefore, the domain (value of t) will be any real number.

b)

Answer: x > -3

Show your work or explain how you obtained your answer here:

Since, the function is of f(x) = ln (x) form, therefore, the value of (x +3) should be greater than 0.

x + 3 > 0 à x > -3

c)

Answer: All real number

Explain how you obtained your answer here:

For , the value of x will be any real number (positive or negative). Therefore, the domain (value of x) will be any real number.

d)

Answer: t > 1

Show your work or explain how you obtained your answer here:

Since, the function is of f(x) = ln (x) form, therefore, the value of (t-1) should be greater than 0.

t-1 > 0 à t > 1

2) Describe the transformations on the following graph of . State the placement of the horizontal asymptote and y-intercept after the transformation. For example, left 1 or rotated about the y-axis are descriptions.

a)

Description of transformation: the curve will move up by 2 point on y-axis.

Horizontal asymptote: y = 2

y-intercept in (x, y) form: (0, 3)

à = 1 + 2 = +1

b)

Description of transformation: The curve will rotate about x-axis (mirror image of ).

Horizontal asymptote: x-axis (y = 0)

y-intercept in (x, y) form: (0, -1)

à = -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, left 1 or stretched vertically by a factor of 2 are descriptions.

a)

Description of transformation: The curve will move right by 3

Vertical asymptote: x = 3

x-intercept in (x, y) form: (4, 0)

à0 = log(x -3)

à x – 3 = 1

à x = 4

b)

Description of transformation: The curve will rotate about y-axis (mirror image of )

Vertical asymptote: x = 0

x-intercept in (x, y) form: (-1, 0)

à0 = log(-x)

à-x = 1

à x = -1

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: the return (A) will be approximately $2938.66

Show work in this space. Use ^ to indicate the power or use the Equation Editor in MS Word.

P = $2,000, t = 5 years, n = 1, and r = 0.08

= = = 2938.656154 ? $2938.66

b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth’s place.

Answer: the return (A) will be approximately $2971.89

Show work in this space:

P = $2,000, t = 5 years, n = 4, and r = 0.08

= = = 2971.894792 ? $2971.89

c) Does compounding annually or quarterly yield more interest? Explain why.

Answer: compounding quarterly yield more interest as compared to compound annually.

Explain:

In case of annual compounding the principal amount remains same for whole years, therefore the interest is computed on this principal amount. However, in case of quarterly compounding, the principal remains same for quarter period (3 months) and interest is added on this principal amount. After that this amount (principal amount + interest) becomes principal amount for next quarter period, therefore, we get more interest for whole year.

For example:

If interest is computed annually for $1 with interest rate = 0.10, than amount after one year will be = 1.11 = $1.10

If Interest is computed quarterly, than = 1.0254 = $1.103813

Therefore, from above example we see that compounding quarterly yield more interest as compared to compound annually.

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: The Amount (A) will be approximately $2983.65

Show work in this space:

P = $2,000, t = 5 years, and r = 0.08

àA = 2000e0.08*5 = 2983.649395 ? $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: It will take approximately 8.66 years (8 years and 8 months) to double the amount.

Show work in this space:

P = $2,000, t = ? years, A = $4000 and r = 0.08

à4000 = 2000 e0.08*t

à e0.08*t = 2 à 0.08 * t = ln 2 à t = (ln 2)/0.08 = 8.66434 ? 8.66 years

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 approximate the percentage of spam in the year 2003.

Answer: The approximate percentage of spam in the year 2003 was 62.44.

Show your work in this space:

à

à P = 13 + 45 * 1.098612 = 13 + 49.437553 = 62.437553

b) Use this model to approximate the year that the percent of spam will reach 95% provided that law enforcement regarding spammers does not change.

Answer: the year that the percent of spam will reach 95% will be 2007.

Show your work in this space:

à 95 = 13 + 45 ln x

à ln x = 82/45 = 1.822222

à x = e1.822222

à x = 6.185589

This means that the year will be 7th i.e. 2007.