A stock analyst plots the price per share of a certain common stock as a function of time and finds that it can be approximated by the function S(t)=46+12e^−0.04t, where t is the time​ (in years) since the stock was purchased. Find the average price of the stock over the first six years.

Answer:

The average price of the stock over the first six years is 56.669 US dollars.

Step-by-step explanation:

Since the price is a continuous and differentiable function, we can determine the average price by means of the following integral equation:

[tex]\bar S = \frac{1}{t_{F}-t_{O}}\cdot \int\limits^{t_{F}}_{t_{O}} {S(t)} \, dt[/tex] (1)

Where:

[tex]t_{O}[/tex] - Initial time, measured in years.

[tex]t_{F}[/tex] - Final time, measured in years.

If we know that [tex]t_{O} = 0\,yr[/tex], [tex]t_{F} = 6\,yr[/tex] and [tex]S(t) = 46+12\cdot e^{-0.04\cdot t}[/tex], then the average price of the stock over the first six years is:

[tex]\bar S = \frac{46}{6-0} \int\limits^6_0 dt +\frac{12}{6-0}\int\limits^6_0 {e^{-0.04\cdot t}} \, dt[/tex]

[tex]\bar S = \frac{23}{3}\cdot t|\limits_{0}^{6}-\frac{2}{0.04}\cdot e^{-0.04\cdot t}|_{0}^{6}[/tex] (2)

[tex]\bar S = \frac{23}{3}\cdot (6-0)-\frac{2}{0.04}\cdot [e^{-0.04\cdot (6)}-1][/tex]

[tex]\bar S = 56.669\,USD[/tex]

The average price of the stock over the first six years is 56.669 US dollars.

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