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Calculus IChapter 3: DerivativesSection 3-2

Interpretation of the Derivative — Question 4

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Question 4

The cost (in dollars) to produce xx items is given by the function: C(x)=50x+300xC(x) = 50x + 300 \sqrt{x}

  • (a) Find the marginal cost function.

  • (b) Interpret the meaning of C′(100)C'(100).

  • (c) Approximate the cost of producing the 101st item using the derivative.

  • (d) Compute the actual cost of producing the 101st item and compare with your approximation.

Original worksheet page 1: question and worked solution for 3-2-004
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Question 4 - Solution

Marginal cost. For x>0x>0,

C′(x)=50+150/x.\boxed{C'(x)=50+150/\sqrt{x}}.

Thus C′(100)=65C'(100)=65 dollars per item. This is the instantaneous cost rate and predicts a cost of approximately $65\boxed{\$65} for one additional item.

Actual additional cost.

C(101)−C(100)=50+300(101−10)≈$64.96.C(101)-C(100)=50+300(\sqrt{101}-10)\approx\boxed{\$64.96}.

The derivative estimate exceeds the actual cost by about $0.04\$0.04. This agrees with C″(x)=−75x−3/2<0C''(x)=-75x^{-3/2}<0: the marginal cost decreases over the next item.

Original worksheet page 2: question and worked solution for 3-2-004

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