Chapter 1.4, Problem 64E

Business: growth of an investment. A company determines that the value of an investment after t years is V, in millions of dollars, where V is given by

$V\left(t\right)=5t^3-30t^2+45t+5\sqrt{t}$.

Note: Calculate often use only the variables y and x, so you may to change the variables.

a. a) Graph V over the interval $\left[0,5\right]$

b. b) Find the equation of the secant line passing through the point $\left(1,V\left(1\right)\right)$ and $\left(5,V\left(5\right)\right)$. Then graph this secant line using the same axes as in part (a).

c. c) Find the average rate of change of the investment between year 1 and year 5.

d. d) Repeat part (b) and (c) for the following pairs of point; $\left(1,V\left(1\right)\right)$

and $\left(4,V\left(4\right)\right)$; $\left(1,V\left(1\right)\right)$ and $\left(3,V\left(3\right)\right)$

$\left(1,V\left(1\right)\right)$ and $\left(1.5,V\left(1.5\right)\right)$.

e. e) What appears to be the slope of the tangent line to the point $\left(1,V\left(1\right)\right)$?

f. Approximate the rate at which the value of the investment is changing at $t=1$ yr.