Question

LOGARITHMIC AND EXPONENTIAL FUNCTIONS.

(ANSWER BASED ON CHAPTER OBJECTIVES: Define an exponential function. Evaluate an exponential expression using the properties of exponents. Simplify an exponential expression using the properties of exponents. Solve an exponential equation. Find the derivative of exponential functions. Define a logarithmic function. Evaluate a logarithmic expression using the properties of logarithms. Simplify a logarithmic expression using the properties of logarithms. Solve a logarithmic equation. Find the derivative of logarithmic functions.)

The two graphs attached show the relationship between the number of items sold per week vs number of weeks since the item arrived at the store. Although both graphs show sales are increasing per week, the graphs look different.

Be sure to respond to both items 1 and 2:

1. Discuss which would be a better investment if you were investing for a period of 5 years. Please provide all details.
   • Option 1: $2000 invested at 6% compounded quarterly
   • Option 2: $2000 invested at 2% compounded continuously
     
2. You just had a baby and want to invest for his college tuition. You found two different accounts:
   In account one you have to invest $2500 at 2% compounded monthly.
   In account two, you have to invest $2000 at 4% compounded annually.
   If you predict that your child will be entering college in 17 years, which account would produce more interest over that time?

Answer 1

Option 1 - $ 2,000 invested at 6% compounded quarterly

Future value = Present value X Future value factor = 2,000 X (1 + 6%/4)^20 = 2,000 X 1.347 = $ 2,694

Option 2 - Continuous compounding

Future value = Present value X e^rt = 2,000 X (2.71828)^0.1 = 2,000 X 1.105171 = $ 2,210.342

Option 1 is far better

In account one,

Future value = 2,500 X (1 + 2%/12)^204 = $ 3,511.375

In account two

Future value = 2,000 X (1 + 4%)^17 = $3,895.801

Option 2 is better

(In case the question meant 2,500 or 2,000 per year, the answer would still be the same that Option 2 is better. Let me know, I will show the calculations)