Calculus Problem

Question

The population p of a small town, in thousands, is given by

p=5⋅1.07^t

where t is time measured in years. Use a calculator or computer to estimate the derivative p′(3)

This means that after 3 years the population is ____ at a rate of ___ people per year.

Answer 1

Hello Gia,

Let's first calculate the derivative of p (p'). This is an exponential equation and the rule is to ignore the coefficient of 5 in the equation and then multiply the term 1.07^t by the natural log of 1.07. Putting these all together yields:

p' = 5 * ln(1.07) * 1.07^t

Plugging in t = 3 gets:

p'(3) = 5* ln(1.07) * 1.07^3 = 0.4144 to four decimal places

To complete the sentence with blanks provided above, the population after three years is your original equation for population with 3 years substituted for t:

p(3) = 5 * 1.07^3 = 6.125 thousands or 6,125 people

To complete the second part, the rate of population growth is p'(3) which, as we already calculated, is equal to 0.4144 thousands per year or 414.4 people per year.