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Unformatted text preview: onential and Logarithmic Functions 3. Our initial investment is $2000, so to ﬁnd the time it takes this to double, we need to ﬁnd t
when A(t) = 4000. We get 2000(1.0059375)12t = 4000, or (1.0059375)12t = 2. Taking natural
logs as in Section 6.3, we get t = 12 ln(1.0059375) ≈ 9.75. Hence, it takes approximately 9 years
9 months for the investment to double.
4. To ﬁnd the average rate of change of A from the end of the fourth year to the end of the
ﬁfth year, we compute A(5)−A(4) ≈ 195.63. Similarly, the average rate of change of A from
the end of the thirty-fourth year to the end of the thirty-ﬁfth year is A(35)−A(34) ≈ 1648.21.
This means that the value of the investment is increasing at a rate of approximately $195.63
per year between the end of the fourth and ﬁfth years, while that rate jumps to $1648.21 per
year between the end of the thirty-fourth and thirty-ﬁfth years. So, not only is it true that
the longer you wait, the more money you have, but also the longer you wait, the faster the
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