Chapter 2, Problem 33RE

(a)

To determine

To find:Themeaning of derivative $f\text{'}\left(r\right)$ and what is its unit.

Answer to Problem 33RE

The meaning of derivative $f\text{'}\left(r\right)$ is the rate of change of the function $C$ , relative to the variable $r$ and the unit is the units of cost divide by the units of interest rate.

Explanation of Solution

Given information:

The given statement is “The total cost of repairing a student loan at an intersect rate of $r\%$ per year is $C=f\left(r\right)$ ”.

Calculation:

The derivative $f\text{'}\left(r\right)$ : The derivative $f\text{'}\left(r\right)$ measures the rate of change of the function $C$ , relative to the variable $r$ . In this case, it is the rate of change of the total cost of repaying the loan, with respect to the interest rate.

The units would be the units of cost divide by the units of interest rate, which would be dollars (percent per year)

Therefore, the meaning of derivative $f\text{'}\left(r\right)$ is the rate of change of the function $C$ , relative to the variable $r$ and the unit is the units of cost divide by the units of interest rate.

(b)

To determine

To find:The meaning of statement “ $f\text{'}\left(10\right)=1200$ ”.

Answer to Problem 33RE

The meaning of statement “ $f\text{'}\left(10\right)=1200$ ” is the interest rate is $10\%$ per year. The additional total cost of repaying the loan is increasing by $\$1200$ for every ear $1\%$ increasing the interest rate.

Explanation of Solution

Given information:

The given statement is “ $f\text{'}\left(10\right)=1200$ ”.

Calculation:

The statement is $f\text{'}\left(10\right)=1200$ means that, when the interest rate is $10\%$ per year. The additional total cost of repaying the loan is increasing by $\$1200$ for every ear $1\%$ increasing the interest rate.

Therefore, the meaning of statement “ $f\text{'}\left(10\right)=1200$ ” is the interest rate is $10\%$ per year. The additional total cost of repaying the loan is increasing by $\$1200$ for every ear $1\%$ increasing the interest rate.

(c)

To determine

To find:Whether $f\text{'}\left(r\right)$ always positive or does it change sign.

Answer to Problem 33RE

The derivative always positive.

Explanation of Solution

Given information:

The given statement is “ $f\text{'}\left(r\right)$ always positive or does it change sign”.

Calculation:

The negative derivative means that, a function is decreasing.

Since higher interest rate will always means a higher total cost of a loan, this function is always increasing.

Therefore, the derivative always positive.