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Subsection 3.4.7 Estimating Change and \(\De x\text{,}\) \(\De y\) Notation

Suppose that we have two variables \(x\) and \(y\) that are related by \(y=f(x)\text{,}\) for some function \(f\text{.}\) One of the most important applications of calculus is to help us understand what happens to \(y\) when we make a small change in \(x\text{.}\)

Definition 3.4.18

Let \(x,y\) be variables related by a function \(f\text{.}\) That is \(y = f(x)\text{.}\) Then we denote a small change in the variable \(x\) by \(\De x\) (read as “delta \(x\)”). The corresponding small change in the variable \(y\) is denoted \(\De y\) (read as “delta \(y\)”).

\begin{align*} \De y &= f(x+\De x) - f(x) \end{align*}

In many situations we do not need to compute \(\De y\) exactly and are instead happy with an approximation. Consider the following example.

Let \(x\) be the number of cars manufactured per week in some factory and let \(y\) the cost of manufacturing those \(x\) cars. Given that the factory currently produces \(a\) cars per week, we would like to estimate the increase in cost if we make a small change in the number of cars produced.

Solution We are told that \(a\) is the number of cars currently produced per week; the cost of production is then \(f(a)\text{.}\)

  • Say the number of cars produced is changed from \(a\) to \(a+\De x\) (where \(\De x\) is some small number.
  • As \(x\) undergoes this change, the costs change from \(y=f(a)\) to \(f(a+\De x)\text{.}\) Hence
    \begin{align*} \De y &= f(a+\De x) - f(a) \end{align*}
  • We can estimate this change using a linear approximation. Substituting \(x=a+\De x\) into the equation 3.4.3 yields the approximation

    \begin{gather*} f(a+\De x)\approx f(a)+f'(a)(a+\De x-a) \end{gather*}

    and consequently the approximation

    \begin{gather*} \De y=f(a+\De x)-f(a)\approx f(a)+f'(a)\De x-f(a) \end{gather*}

    simplifies to the following neat estimate of \(\De y\text{:}\)

  • In the automobile manufacturing example, when the production level is \(a\) cars per week, increasing the production level by \(\De x\) will cost approximately \(f'(a)\De x\text{.}\) The additional cost per additional car, \(f'(a)\text{,}\) is called the “marginal cost” of a car.
  • If we instead use the quadratic approximation (given by equation 3.4.6) then we estimate

    \begin{gather*} f(a+\De x)\approx f(a)+f'(a)\De x+\half f''(a)\De x^2 \end{gather*}

    and so

    \begin{align*} \De y&=f(a+\De x)-f(a) \approx f(a)+f'(a)\De x +\half f''(a)\De x^2-f(a) \end{align*}

    which simplifies to