Single Variable Calculus, Chapter 3, Review Exercises, Section Review Exercises, Problem 76

The equation $C(x) = 920 + 2x - 0.02x^2 + 0.0007 x^3$ represents the cost, in dollars, of producing $x$ units of certain commodity.

a.) Determine the marginal cost function


$
\begin{equation}
\begin{aligned}

& \text{marginal cost function }
\\
\\
C'(x) =& \frac{d}{dx} 920 + 2 \frac{d}{dx} (x) - 0.02 \frac{d}{dx} (x^2) + 0.00007 \frac{d}{dx} (x^3)
\\
\\
C(x) =& 0 + 2(1) - 0.02 (2x) + 0.00007 (3x^2)
\\
\\
C(x) =& 2 - 0.04 x + 0.00021 x^2

\end{aligned}
\end{equation}
$



b.) Find $C'(100)$ and explain its meaning.


$
\begin{equation}
\begin{aligned}

C'(100) =& 2 - 0.04 (100) + 0.00021 (100)^2
\\
\\
C'(100) =& \frac{1}{10 } \text{ dollars}

\end{aligned}
\end{equation}
$


$C'(100)$ represents the change in cath with respect to the amount produced. $C'(100)$ predicts how much one extra unit would costs after $100$ units have already been produced.

c.) Compare $C'(100)$ with the cost of producing the $101st$ item

The cost of producing the $101st$ term = $C(101) - C(100)$


$
\begin{equation}
\begin{aligned}

=& 920 + 2 (101) - 0.02(101)^2 + 0.00007 (101)^3 - [920 + 2 (100) - 0.02(100)^2 + 0.00007 (100)^3]
\\
\\
=& 0.10107

\end{aligned}
\end{equation}
$


We can see that $\displaystyle C'(100) = \frac{1}{10}$ and $0.10107$ are nearly the same.