Chapter 6 | Applications of Integration
737
nt
1000 lim n →∞ ⎛
⎞ ⎠
⎝ 1+ 0.02 n . Now let’s manipulate this expression so that we have an exponential growth function. Recall that the number e can be expressed as a limit: e = lim m →∞ ⎛ ⎝ 1+ 1 m ⎞ ⎠ m . Based on this, we want the expression inside the parentheses to have the form (1+1/ m ). Let n =0.02 m . Note that as n →∞, m →∞ as well. Then we get . We recognize the limit inside the brackets as the number e . So, the balance in our bank account after t years is given by 1000 e 0.02 t . Generalizing this concept, we see that if a bank account with an initial balance of $ P earns interest at a rate of r %, compounded continuously, then the balance of the account after t years is Balance= Pe rt . Example 6.43 Compound Interest 1000 lim n →∞ ⎛ ⎝ 1+ 0.02 n ⎞ ⎠ nt =1000 lim m →∞ ⎛ ⎝ 1+ 0.02 0.02 m ⎞ ⎠ 0.02 mt =1000 ⎡ ⎣ lim m →∞ ⎛ ⎝ 1+ 1 m ⎞ ⎠ m ⎤ ⎦ 0.02 t A 25-year-old student is offered an opportunity to invest some money in a retirement account that pays 5% annual interest compounded continuously. How much does the student need to invest today to have $1 million when she retires at age 65? What if she could earn 6% annual interest compounded continuously instead?
Solution We have
1,000,000 = Pe 0.05(40) P = 135,335.28.
She must invest $135,335.28 at 5% interest. If, instead, she is able to earn 6%, then the equation becomes
1,000,000 = Pe 0.06(40) P = 90,717.95.
In this case, she needs to invest only $90,717.95. This is roughly two-thirds the amount she needs to invest at 5%. The fact that the interest is compounded continuously greatly magnifies the effect of the 1% increase in interest rate.
6.43 Suppose instead of investing at age 25 , the student waits until age 35. How much would she have to invest at 5%? At 6%?
If a quantity grows exponentially, the time it takes for the quantity to double remains constant. In other words, it takes the same amount of time for a population of bacteria to grow from 100 to 200 bacteria as it does to grow from 10,000 to 20,000 bacteria. This time is called the doubling time. To calculate the doubling time, we want to know when the quantity reaches twice its original size. So we have
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