Questions: Find the marginal cost function. C(x)=240+7.9 x-0.07 x^2 C'(x)=

Transcript text: Find the marginal cost function. \[ \begin{array}{l} C(x)=240+7.9 x-0.07 x^{2} \\ C^{\prime}(x)=\square \end{array} \]

Solution

Solution Steps

Step 1: Understand the Given Total Cost Function

The given total cost function is in the form $C(x) = a + bx - cx^2$, where:

$a$ represents the fixed cost, which is {a}.
$b$ represents the variable cost coefficient, which is {b}.
$c$ represents the coefficient for the quadratic term, which is {c}.

In this scenario, $x$ represents the quantity of goods produced.

Step 2: Derive the Marginal Cost Function

The marginal cost function, $C'(x)$, is found by taking the first derivative of the total cost function with respect to $x$. Thus, $$C'(x) = \frac{d}{dx}(a + bx - cx^2) = b - 2cx$$

Step 3: Calculate the Marginal Cost

Substituting the given values into the marginal cost function, we get:

$$C'(x) = 7.9 - 2\cdot0.07\cdot50 = 0.9$$

Final Answer:

The marginal cost of producing 50 units is approximately $0.9 (rounded to 2 decimal places).

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