Question

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 1

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

Explanation:

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

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

Where:

`t_(O)` - Initial time, measured in years.

`t_(F)` - Final time, measured in years.

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

`\bar S = (46)/(6-0) \int\limits^6_0 dt +(12)/(6-0)\int\limits^6_0 {e^(-0.04\cdot t)} \, dt`

`\bar S = (23)/(3)\cdot t|\limits_(0)^(6)-(2)/(0.04)\cdot e^(-0.04\cdot t)|_(0)^(6)` (2)

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

`\bar S = 56.669\,USD`

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