Question

a company’s marginal cost function is $c(x)=\frac{20}{sqrt{x}}$ where $x$ is the number of units. find the total cost of the first 64 units (from $x = 0$ to $x = 64$) total cost: $

Explanation:

Step1: Rewrite marginal cost function

Rewrite $c(x)=\frac{20}{\sqrt{x}}$ as $c(x)=20x^{-\frac{1}{2}}$

Step2: Set up total cost integral

Total cost is the definite integral of marginal cost from 0 to 64:
$$\int_{0}^{64} 20x^{-\frac{1}{2}} dx$$

Step3: Apply power rule for integration

Power rule: $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ ($n
eq-1$)
$$\int 20x^{-\frac{1}{2}} dx = 20\cdot\frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C = 40\sqrt{x} + C$$

Step4: Evaluate definite integral

Substitute bounds 64 and 0:
$$40\sqrt{64} - 40\sqrt{0} = 40\cdot8 - 0$$

Answer:

$\$320$