Exemplo 7 de Precalculus, 12.4 Derivatives

Finding a Marginal Cost

Problema e solução em inglês, como no livro original.

The cost in dollars of producing $x$ laptop computers in dollars is $f (x) = x^{2} - 100 x .$ At the point where 200 computers have been produced, what is the approximate cost of producing the 201^st unit?

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

Solução
If $f (x) = x^{2} - 100 x$ describes the cost of producing $x$ computers, $f^{'} (x)$ will describe the marginal cost. We need to find the derivative. For purposes of calculating the derivative, we can use the following functions:

$\begin{array}{l} f (a + b) = {(x + h)}^{2} - 100 (x + h) \\ f (a) = a^{2} - 100 a \end{array}$
$\begin{array}{l} f^{'} (x) = \frac{f (a + h) - f (a)}{h} & Formula for a derivative \\ = \frac{{(x + h)}^{2} - 100 (x + h) - (x^{2} - 100 x)}{h} & Substitute f (a + h) and f (a) . \\ = \frac{x^{2} + 2 x h + h^{2} - 100 x - 100 h - x^{2} + 100 x}{h} & Multiply polynomials, distribute . \\ = \frac{2 x h + h^{2} - 100 h}{h} & Collect like terms . \\ = \frac{h (2 x + h - 100)}{h} & Factor and cancel like terms . \\ = 2 x + h - 100 & Simplify . \\ = 2 x - 100 & Evaluate when h = 0. \\ f^{'} (x) = 2 x - 100 & Formula for marginal cost \\ f^{'} (200) = 2 (200) - 100 = 300 & Evaluate for 200 units . \end{array}$
The marginal cost of producing the 201^st unit will be approximately $300.

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