Rates of Change in Applied Contexts Practice

Sample questions

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1. Sample 1difficulty 2/5

   If $C(100) = 5000$ and $C'(100) = 12$, the cost of producing the 101st unit is approximately

   • A

     $5012$

   • B

     $12$

   • C

     $5000$

   • D

     \5012$— but the marginal cost ($12$) approximates the additional cost

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   Why

   Marginal cost $C'(100) = 12$ approximates the cost of unit 101.

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2. Sample 2difficulty 2/5

   If $C(x)$ is total cost for $x$ items, "marginal cost" means

   • A

     $C'(x)$

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   • B

     $\int C(x)\,dx$

   • C

     $C(x)$

   • D

     $C(x)/x$

   Why

   Marginal cost = derivative of cost = $C'(x) \approx$ cost of the next item.

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3. Sample 3difficulty 2/5

   If $R(x) = x(100 - 2x)$ is revenue from selling $x$ units, marginal revenue at $x = 20$ is

   • A

     $60$

   • B

     $80$

   • C

     $0$

   • D

     $20$

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   Why

   $R(x) = 100x - 2x^2$. $R'(x) = 100 - 4x$. At $x = 20$: $20$.

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4. Sample 4difficulty 2/5

   If $P(x) = R(x) - C(x)$ is profit, profit is maximized when

   • A

     $x = 0$

   • B

     $R'(x) = C'(x)$

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   • C

     $R(x) = C(x)$

   • D

     $P(x) = 0$

   Why

   $P'(x) = 0 \Rightarrow R'(x) = C'(x)$ — marginal revenue equals marginal cost.

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