### Subsection3.4.7Estimating 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 $x\text{.}$

###### Definition3.4.18

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