Percentage rate of growth

Question

Suppose that the value of a company, in millions of dollars, is determined to be modeled by the function f(t)=0.9t+0.4e^-2t, where t is measured in years. To the nearest tenth, what is the percentage rate of growth of the company at time t=2?

Answer 1

We are looking for the rate of growth (change) in the company, which means we need the derivative of the value of the company.

f(t) = 0.9t + 0.4e^(-2t)

Take the derivative of the function.

With the 0.4e^(-2t) term, the exponent stays the same when taking the derivative, and the -2 is multiplied to the 0.4 constant.

We have:

f'(t) = 0.9 + (0.4)(-2)e^(-2t)

f'(t) = 0.9 - 0.8e^(-2t)

Now that we have the derivative, we can find the rate of growth at year t = 2 by plugging into the new equation.

f'(2) = 0.9 - 0.8e^(-2(2))

f'(2) = 0.9 - 0.01465

f'(2) = 0.88535

We multiply this result by 100 to find the percentage as that is what is being asked.

= 88.535%

Round to the nearest tenth

= 88.5%

This is the solution.