Derivatives and rates of change

Slopes of lines and tangent lines

  • Recall slopes
    • Let \(m\) be a number, and \(P(a,c)\) a plane point.
    • Then the equation of the line through \(P\) with slope \(m\) is the following. \[ y-c=m(x-a) \]
    • How you can remember this is that \(y=mx\) is the line with slope \(m\) through the origin \(O(0,0)\), which you then need to translate by \(c\) upwards, and by \(a\) to the right.
  • Tangent lines
    • Let \(f(x)\) be a function, and let \(C:y=f(x)\) be the curve which is its graph.
    • Let us fix two numbers \(a\) and \(b\) such that the closed interval \([a,b]\) is contained in the domain of \(f(x)\), and consider the points \(P(a,f(a))\) and \(Q(b,f(b))\) on \(C\).
    • Then the line between \(P\) and \(Q\) will have the following slope. \[ m_{PQ}=\frac{f(b)-f(a)}{b-a}. \]
    • Definition. Let \(f(x)\) be a function, and \(a\) a point in its domain. Then the tangent line to the curve \(C=f(x)\) at \(P(a,f(a))\) is the line through \(P\) with slope \[ m=\lim_{b\to a}\frac{f(b)-f(a)}{b-a}, \] provided that the limit exists.
    • Click here for an illustration.
    • Exercise. 2.7: 4aib. Click here for part c

Rate of change and derivatives

  • Let \(f(x)\) be a function, and \(a\) and \(b\) two numbers such that the closed interval \([a,b]\) is contained in the domain of \(f(x)\).
    • We can interpret the difference \(f(b)-f(a)\) as the average rate of change of \(f(x)\) on the interval \([a,b]\).
    • There's another notation emphasising this point of view.
    • The change in \(x\) is \(\Delta x=b-a\), and the change in \(y=f(x)\) is \(\Delta y=f(b)-f(a)\).
    • Then the average rate of change can be written as \(\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}\).
  • As we make \(b\to a\), we get the (instantaneous) rate of change at \(x=a\).
    • Since we make \(b\to a\), it's better to write \(b=a+h\).
    • Note that \(h=\Delta x=b-a\).
    • Definition. Let \(f(x)\) be a function, and \(a\) a number such that the domain of \(f(x)\) contains some open interval around \(a\). Then the derivative of \(f(x)\) at \(x=a\) is \[ f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}, \] provided that the limit exists.
    • An alternative notation is \(\frac{\mathrm df}{\mathrm dx}\big|_{x=a}\).
    • Exercises. 2.7: 8ab, 10ab, 14, 20, 22, 60