what is the rate of change?

Question

Suppose the population of a city can be modeled by P(t)=2130e0.13tP(t)=2130e0.13t persons, where tt is the number of years since 2000.

Meanwhile, the number of college graduates in the city can be modeled by D(t)=150+117tD(t)=150+117t, where tt is the number of years since 2000.

A function giving the fraction of college graduates is R(t)=R(t)=   .

In 2001, the fraction of college graduates in the city is 

1. increasing
2. decreasing

at a rate of    per year (round your answer to 3 decimal places)

Answer 1

Two variables must be more precisely defined here.

Population of the city since the year 2000: P(t) = 2130e^.13t

Number of college graduates in the city since the year 2000: D(t) = 150 + 117t

The fraction of college graduates in the city is represented as R(t) = D(t) / P(t).

To find how R(t) changes requires us to derive D(t) / P(t). In this case, the rate of change of the fraction of college graduates in the city is (d/dt)[ D(t) / P(t) ]. We are deriving with respect to time because D(t) and P(t) are with respect to time.

Quotient rule will be needed to derive D(t) / P(t). (d/dt)[ D(t) / P(t) ] will be equivalent to...

---> [ (D^'(t) * P(t)) - (D(t) * P^'(t)) ] / (P(t))². Bear in mind that the (P(t))² divides everything else.

Here, D^'(t) = 117 and P^'(t) = (.13)(2130e^.13t). D^'(t) was found using the constant and power rules while the natural exponent rule was used to derive P^'(t). More information below...

Constant rule: (d/du)[ c ] = 0 where c is a constant and u is any integration variable.

Power Rule: (d/du)[ uⁿ ] = nu^n-1 where u is any integration variable and n is any real number.

Natural exponent rule: (d/du)[ e^u ] = u^' * e^u where u^' is the derivative of u and u is an expression (like u² or sin(u)).

Using the derivatives of D(t) and P(t), substituting back these values, and defining the derivative as R^'(t)...

(d/dt)[ D(t) / P(t) ] = R^'(t) = [ (117) * (2130e^.13t)) - ((150 + 117t) * ((.13)2130e^.13t)) ] / (2130e^.13t)²

Now we can answer the questions.

It is easier to answer question two first because that value will answer number one.

Bear in mind that these equations take place since 2000. Thus 2001 will translate to R^'(1).

Using a calculator, R^'(1) = .03392 to five decimal places. This is our answer to question two.

Question one depends on the sign of R^'(1). Since R^'(1) > 0, R(t) is increasing. This is our answer to question one.