« Section 6.2
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Section 6.3

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1. (a) \displaystyle{\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e}

(c) \displaystyle{\lim_{n \to \infty}\left(1 - \frac{2}{n}\right) = e^{-2}}

(e) \displaystyle{\lim_{n \to \infty}\left(1 + \frac{2}{n^2}\right) = 1}

(f) \displaystyle{\lim_{n \to \infty}\left(1 - \frac{4}{n} + \frac{1}{n^2}\right) = e^{-4}}

3. (a) \$6946.20

(b) \$6961.83

(c) \$6967.91

(d) \$6969.48

(e) \$6969.74

4. The 5.5 % interest compounded quarterly is the most advantageous.

6. 24,765 years

7. The bone is between 23,257 years and 26,609 years old.

8. x(t) = x_0e^{-0.034657t}
66.4 minutes until 10% remains; 86.4 minutes until 5% remains

9. x(t) = x_0e^{-0.000028881t}
79,726 years until 10% remains; 103,726 years until 5% remains

10. 69 years

11. (a) y(t) = 179.3e^{0.012562t}

(b) Prediction for 1980: 230.5 million
Prediction for 1990: 261.4 million
Prediction for 2000: 296.3 million

13. (a) x(t) = 75.995e^{0.0190825t} , in millions
Prediction for 1920: 111.3 million
Prediction for 1930: 134.7 million
Prediction for 1950: 197.3 million
Prediction for 1970: 289.0 million
Prediction for 1990: 423.3 million
Prediction for 2000: 512.3 million
The population in 1936 would be twice the population of 1900.

(b) \displaystyle{x(t) = \frac{10595}{75.995 + 63.424e^{-0.048106t}}}, in millions
Prediction for 1930: 116.5 million
Prediction for 1950: 129.7 million
Prediction for 1970: 135.5 million
Prediction for 1990: 137.9 million
Prediction for 2000: 138.5 million
This model predicts a limiting population of M = 139.4 million. Since this is less than twice the population in 1900, this model predicts that the population will never double from its 1900 level.

14. Hint: \displaystyle{\ddot{x}(t) = \alpha \dot{x}(t) - \frac{2\alpha}{M}x(t)\dot{x}(t)}
This tells us that the rate of growth of the population is increasing when the population is less than one-half of its limiting size and decreasing when the population exceeds one-half of its limiting size.

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