Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 27

The equation $c(x) = 1200 + 12x - 0.1x^2 + 0.0005 x^3$ represents the cost, in dollars, of producing $x$ yards of a certain fabric.
a.) Find the merginal cost function.
b.) Find $c'(200)$ and explain its meaning. What does it predict?
c.) Compare $c'(200)$ with the cost of manufacturing the 201st yard of fabric.

a.) The marginal cost function $c'(x)$ is...

$
\begin{equation}
\begin{aligned}
c'(x) &= \frac{d}{dx} (1200) + 12 \frac{d}{dx} (x) - 0.1 \frac{d}{dx} (x^2) + 0.0005 \frac{d}{dx} (y^3)\\
\\
c'(x) &= 0 + 12(1) - 0.1(2x) + 0.0005(3x^2)\\
\\
c'(x) &= 12 - 0.2x + 0.0015x^2
\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}
\text{b.) } c'(200) &= 12 - 0.2(200) + 0.0015(200)^2\\
\\
c'(200) &= 32 \frac{\text{dollars}}{\text{yards}}

\end{aligned}
\end{equation}
$

$c'(200)$ represents the incremental cost to produce one more yard of fabric after producing 200 yards of fabric. The cost of manufacturing the 201st yard should be \$32.


$
\begin{equation}
\begin{aligned}
\text{c.) } c(201) - c(200) &= 1200 + 12(201) - 0.1(201)^2 + 0.0005(201)^3 - \left[ 1200 + 12(200) - 0.1 (200)^2 + 0.0005(200)^3 \right]\\
\\
&= 32.2005 \text{ dollars}
\end{aligned}
\end{equation}
$

The answer of \$32.2005 is somewhat exactly what we predict in part (b).