Section 3.1 Defining the Derivative
References.
- Calculus, Early Transcendentals by Stewart, Section 2.7.
This section revisits ideas seen earlier in Section 2.1, now done more completely and efficiently using what we know about limits, and revisits examples from those sections. Thus we will not work all the examples in class, but I recommend that you read the whole section and study all the examples.
Tangents.
In the Preview of Calculus we saw that the slope of the secant line on the curve \(y=f(x)\) between a point \(P(a,f(a))\) and another point \(Q(x,f(x))\text{,}\) \(x \neq a\text{,}\) is
\begin{equation*}
m_{PQ} = \frac{\Delta y}{\Delta x} = \frac{f(x)-f(a)}{x-a}.
\end{equation*}
This “slope function” \(\frac{f(x)-f(a)}{x-a}\) is undefined at \(x=a\text{,}\) but often it has a removable discontinuity there. We also saw there that it makes sense to define the slope of the curve at \(P\) as the limit of this secant slope as \(x \to a\text{:}\)
\begin{equation*}
m = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}.
\end{equation*}
Typically, the line of this slope \(m\) through \(P\) touches the curve but does not cross it, so we call it the tangent line to \(y=f(x)\) at point \(P(a,f(a))\), or the tangent at \(x=a\).
It is often convenient to let \(h=x-a\text{,}\) the horizontal increment, so that \(x=a+h\) and the tangent slope is given by
\begin{equation*}
m = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}
\end{equation*}
This makes it easier to identify the division by zero that we wish to eliminate in order to compute the limit. The formulation in terms of step size \(h\) is even more useful when the algebra gets more complicated.
Find the slopes of the tangent line to \(y=f(x) = \sqrt{x}\) at the points \((1,1)\text{,}\) \((4,2)\) and \((9,3)\text{,}\) by computing it at a general point \(P(a,\sqrt{a})\text{.}\)
Hint.Use the “rationalizing factor”
\begin{equation*}
1 = \frac{\sqrt{a+h}+\sqrt{a}}{\sqrt{a+h}+\sqrt{a}}.
\end{equation*}
Velocities.
In the Preview we saw that average velocity is given by a formula like that for secant slope. For an object whose position at time \(t\) is \(f(t)\text{,}\) the average velocity over a time interval of duration \(h\) from time \(a\) to time \(a+h\) is
\begin{equation*}
v_{ave} = \frac{f(a+h)-f(a)}{h}.
\end{equation*}
The instantaneous velocity at time \(\mathbf{a}\) is the limit of this as the length of the time interval \(h\) approaches zero:
\begin{equation*}
v(a) = \lim_{h \to 0}\frac{f(a+h)-f(a)}{h}.
\end{equation*}
The quantity \(\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\) has been seen to be important to computing slopes, velocities and other rates of change. It deserve a name, and a short-hand, \(f'(a)\text{:}\)
Definition 3.1.2.
The derivative of function \(f\) at number \(a\) is the quantity
\begin{equation*}
f'(a) = \lim_{h \to 0}\frac{f(a+h)-f(a)}{h},% \; = \lim_{x \to a}\frac{f(x)-f(a)}{x-a},
\end{equation*}
if this limit exists. An alternative form is
\begin{equation*}
f'(a) = \lim_{x \to a}\frac{f(x)-f(a)}{x-a}, \text{ where } x = a+h.
\end{equation*}
Interpretation of the Derivative as the Slope of a Tangent Line.
The quantity now called \(f'(a)\) has been seen as the slope \(m\) of the tangent line to a curve. We can now write that
Definition 3.1.3.
The tangent line to curve \(y=f(x)\) at point \(P(a,f(a))\) is the line with equation
\begin{equation*}
y = l(x) = f(a)+f'(a)(x-a)
\end{equation*}
Note well that only \(x\) is the variable in the function \(l(x)\) here: \(a\) is a constant, and so \(f(a)\) and \(f'(a)\) are also constants.
Interpretation of the Derivative as a Rate of Change.
The derivative of position as a function of time is velocity, or the (time) rate of change of position. Likewise the derivative of a function is the rate of change of the value of the function value with respect to change in the value of its argument.
For any quantity \(y\) related to another quantity \(x\) by \(y=f(x)\text{,}\) changing the value of \(x\) from \(x_1\) to \(x_2\) causes a change in \(y\) from \(y_1=f(x_1)\) to \(y_2=f(x_2)\text{,}\) so that the change by \(\Delta x=x_2-x_1\) in \(x\) causes a change of by \(\Delta y=y_2-y_1\) in \(y\text{.}\) The difference quotient, defined by
\begin{equation*}
\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2-x_1}
\end{equation*}
gives the average rate of change of \(y\) respect to \(x\) over the interval \([x_1,x_2]\text{.}\)
As we adjust \(x_2\) to approach \(x_1\text{,}\) so that \(\Delta x\) approaches 0, this average rate of change approaches the instantaneous rate of change of \(y\) with respect to \(x\text{,}\)
\begin{equation*}
\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x},\, = \lim_{x_2 \to x_1}\frac{f(x_2) - f(x_1)}{x_2-x_1}.
\end{equation*}
With some name changes (\(x_1\) becoming \(a\text{;}\) \(x_2\) becoming \(a+h\text{,}\) so that \(\Delta x\) becomes \(h\)) this is the same as the definition of the derivative:
The derivative \(f'(a)\) of function \(f\) at number \(a\) is the instantaneous rate of change of \(y=f(x)\) with respect to \(x\) when \(x=a\text{.}\)
Exercises Exercises
Study Calculus Volume 1, Section 3.1 2 ; in particular Examples 1,2, 3, 5, 6 and 9, Checkpoint items 1, 3 and 4, and Exercises 1, 7, 11, 13, 15, 25, 37, 39, 41 and 51.
openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivativeopenstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative
