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Calculus/Precalculus Questions

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Expert: - 10/19/2009


Question
I have some difficult questions, could you show me how you get the answer?

1. Tashi deposits $10,000 into a retirement account and leaves it to accumulate for 25 years. Find the accumulated amount of the interest rate is (a) a true annual rate of 8 percent, (b) 8 percent compounded monthly, (c) 8 percent compounded daily, and (d) 8 percent compounded continuously.

2. Advanced calculus students know that ex can be expanded in a Maclaurin series to obtain ex=1+x+(x^2/2!)+(x^3/3!)+(x^4/4!)+...
Let x=1 and obtain an approximation for e. How many terms of this series are needed to determine e accurate to two decimal places? To three decimal places?

Answer

Approximating e
1a) The amount is $10,000, the interest is 8% = 0.08, and the years are 25.
The final amount is F = $10,000(1.08)^25.

1b) If 8% is compounded monthly, that is 8%/12 per month.
The final result would then be F = $10,000(1 + 0.2/3)^(25*12), where 25*12 = 300
and 1 + 0.02/3 = 1.006666666....

1c) Most bankers use to assume there were 360 days in a year, so it would be
$10,000(1 + 0.08/360)^9000.

1d) $10,000*e^(0.08*25) = %10,000*e^2, where e = 2.71828182845905....

2) 4 terms will tell you what e is to two places, 5 to three decimal places.

Scotto

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