# C(x)=5,000+20x-1/4x^2 is a cost function. Calculate the marginal

$$\displaystyle{C}{\left({x}\right)}={5},{000}+{20}{x}-\frac{{1}}{{4}}{x}^{{2}}$$
is a cost function.
Calculate the marginal average cost when 20 (a), use your result to estimate the total cost when x=21 (b) and calculate the percentage error between this estimate and the actual cost when x=21 (c).

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lamusesamuset
$$\displaystyle{C}{\left({x}\right)}={5},{000}+{20}{x}-\frac{{1}}{{4}}{x}^{{2}}$$
$$\displaystyle\frac{{{C}{\left({x}\right)}}}{{x}}=?$$
$$\displaystyle\frac{{{C}{\left({x}\right)}}}{{x}}$$
$$\displaystyle=\frac{{{5},{000}+{20}{x}-\frac{{1}}{{4}}{x}^{{2}}}}{{x}}$$
$$\displaystyle=\frac{{5000}}{{x}}+{20}-\frac{{1}}{{4}}{x}$$
$$\displaystyle\overline{{C}}'{\left({x}\right)}=\frac{{{d}{\left(\frac{{5000}}{{x}}+{20}-\frac{{1}}{{4}}{x}\right)}}}{{{\left.{d}{x}\right.}}}$$
$$\displaystyle=-\frac{{5000}}{{x}^{{2}}}-\frac{{1}}{{4}}$$
So $$\displaystyle\overline{{C}}'{\left({20}\right)}=-\frac{{5000}}{{x}^{{2}}}-\frac{{1}}{{4}}$$
$$\displaystyle=-{12.5}-{0.25}$$
$$\displaystyle=-{12.75}$$
And $$\displaystyle{C}{\left({20}\right)}=\frac{{5000}}{{20}}+{20}-\frac{{1}}{{4}}{\left({20}\right)}={265}$$ (a)
$$\displaystyle\overline{{C}}{\left({21}\right)}=\overline{{C}}{\left({20}\right)}+\overline{{C}}'{\left({20}\right)}$$
$$\displaystyle={265}-{12.75}$$
$$\displaystyle={252.25}$$
$$\displaystyle{C}\frac{{{21}}}{{21}}={252.25}$$
$$\displaystyle{C}{\left({21}\right)}={5297.25}$$ (b)
$$\displaystyle{\underset{{{E}}}{{{C}}}}={5000}+{20}{\left({21}\right)}-\frac{{1}}{{4}}{\left({21}\right)}^{{2}}$$
$$\displaystyle={5309.75}$$
$$\displaystyle\frac{{{5309.75}-{5297.25}}}{{5309.75}}\times{100}={0.24}\%$$ (c)