Skip to main content
\(\require{cancel}\newcommand{\half}{ \frac{1}{2} } \newcommand{\ds}{\displaystyle} \newcommand{\ts}{\textstyle} \newcommand{\es}{ {\varnothing}} \newcommand{\st}{ {\mbox{ s.t. }} } \newcommand{\pow}[1]{ \mathcal{P}\left(#1\right) } \newcommand{\set}[1]{ \left\{#1\right\} } \newcommand{\lin}{{\text{LIN}}} \newcommand{\quot}{{\text{QR}}} \newcommand{\simp}{{\text{SMP}}} \newcommand{\diff}[2]{ \frac{\mathrm{d}#1}{\mathrm{d}#2}} \newcommand{\bdiff}[2]{ \frac{\mathrm{d}}{\mathrm{d}#2} \left( #1 \right)} \newcommand{\ddiff}[3]{ \frac{\mathrm{d}^#1#2}{\mathrm{d}{#3}^#1}} \renewcommand{\neg}{ {\sim} } \newcommand{\limp}{ {\;\Rightarrow\;} } \newcommand{\nimp}{ {\;\not\Rightarrow\;} } \newcommand{\liff}{ {\;\Leftrightarrow\;} } \newcommand{\niff}{ {\;\not\Leftrightarrow\;} } \newcommand{\De}{\Delta} \newcommand{\bbbr}{\mathbb{R}} \newcommand{\arccsc}{\mathop{\mathrm{arccsc}}} \newcommand{\arcsec}{\mathop{\mathrm{arcsec}}} \newcommand{\arccot}{\mathop{\mathrm{arccot}}} \newcommand{\erf}{\mathop{\mathrm{erf}}} \newcommand{\smsum}{\mathop{{\ts \sum}}} \newcommand{\atp}[2]{ \genfrac{}{}{0in}{}{#1}{#2} } \newcommand{\YEaxis}[2]{\draw[help lines] (-#1,0)--(#1,0) node[right]{$x$};\draw[help lines] (0,-#2)--(0,#2) node[above]{$y$};} \newcommand{\YEaaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$x$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YEtaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$t$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YExcoord}[2]{\draw (#1,.2)--(#1,-.2) node[below]{$#2$};} \newcommand{\YEycoord}[2]{\draw (.2,#1)--(-.2,#1) node[left]{$#2$};} \renewcommand{\textcolor}[2]{\color{#1}{#2}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Chapter3Applications of derivatives

In Section 2.2 we defined the derivative at \(x=a\text{,}\) \(f'(a)\text{,}\) of an abstract function \(f(x)\text{,}\) to be its instantaneous rate of change at \(x=a\text{:}\)

\begin{align*} f'(a) &= \lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a} \end{align*}

This abstract definition, and the whole theory that we have developed to deal with it, turns out be extremely useful simply because “instantaneous rate of change” appears in a huge number of settings. Here are a few examples.

  • If you are moving along a line and \(x(t)\) is your position on the line at time \(t\text{,}\) then your rate of change of position, \(x'(t)\text{,}\) is your velocity. If, instead, \(v(t)\) is your velocity at time \(t\text{,}\) then your rate of change of velocity, \(v'(t)\text{,}\) is your acceleration. We shall explore this further in Section 3.1.
  • If \(P(t)\) is the size of some population (say the number of humans on the earth) at time \(t\text{,}\) then \(P'(t)\) is the rate at which the size of that population is changing. It is called the net birth rate. We shall explore it further in Section 3.3.3.
  • Radiocarbon dating, a procedure used to determine the age of, for example, archaeological materials, is based on an understanding of the rate at which an unstable isotope of carbon decays. We shall look at this procedure in Section 3.3.1
  • A capacitor is an electrical component that is used to repeatedly store and release electrical charge (say electrons) in an electronic circuit. If \(Q(t)\) is the charge on a capacitor at time \(t\text{,}\) then \(Q'(t)\) is the instantaneous rate at which charge is flowing into the capacitor. That's called the current. The standard unit of charge is the coulomb. One coulomb is the magnitude of the charge of approximately \(6.241 \times 10^{18}\) electrons. The standard unit for current is the amp. One amp represents one coulomb per second.