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4.1 Secant lines

This section introduces the derivative. Recall derivative measures change, change is what is of interest, so we are measuring what is of interest!

Here's a sketch of how we will get to the derivative:

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Draw a picture of the secant line approximation to a function at a particular point (x).
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The tangent line is the limit of the secant line as the "interval" ($\Delta x$) gets smaller.
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The derivative is the slope of the tangent line at x.
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The derivative of a function at the point x is defined as

\begin{displaymath}\lim_{\Delta x\rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.\end{displaymath}

In English, the derivative is the "rate of change" of the function.

Take a look at these ideas pictorially:

As $\Delta x$ gets smaller and smaller, how much does y change, and what about the slope, that is the ratio of $\frac{\Delta y}
{\Delta x}$?


 
Table 1: Calculation of the slope of the secant line approximation to the function y = x2 about the point x = 3.
x y = x2 $\Delta x$ $x + \Delta x$ $y = f(x + \Delta x)$ $\Delta y$ $\frac{\Delta y}
{\Delta x}$
3 9 4 7 49 40 10
3 9 3 6 36 27 9
3 9 2 5 25 16 8
3 9 1 4 16 7 7
3.00 9.00 0.50 3.50 12.25 3.25 6.50
3.0000 9.0000 0.2500 3.2500 10.5625 1.5625 6.2500
3.00 9.00 0.10 3.10 9.61 0.61 6.10
3.0000 9.0000 0.0100 3.0100 9.0601 0.0601 6.0100

Note, that the ratio tends to a limit!

Call this limit, $\frac{dy}{dx}$. Note the lower case d's rather than the $\Delta$'s.

Formally defined as


\begin{displaymath}\lim_{\Delta x\rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.\end{displaymath}

Do an example (once) ...

Take the function, f, where f(x) = x2.


\begin{displaymath}\lim_{\Delta x \rightarrow 0} \frac{f(3 + \Delta x) - f(3)}{\Delta x}.\end{displaymath}


\begin{displaymath}\lim_{\Delta x \rightarrow 0} \frac{(3 + \Delta x)^2 - 3^2}{\Delta x}.\end{displaymath}


\begin{displaymath}\lim_{\Delta x \rightarrow 0} \frac{9 + 6\,\Delta x + (\Delta x)^2 - 9}{\Delta x}.\end{displaymath}


\begin{displaymath}\lim_{\Delta x \rightarrow 0} \frac{6\,\Delta x + (\Delta x)^2}{\Delta x}.\end{displaymath}

'

\begin{displaymath}\lim_{\Delta x \rightarrow 0} (6 + \Delta x) = 6.\end{displaymath}

So, the answer is 6, just as the picture suggested.

Back to interpretation.

Consider a function that tells you how far you have gone after a particular amount of time. For example, think about flying from JFK to LHR; how far have you traveled in 4.5 hours etc.

Call x the time measured in hours, and y the distance you have traveled measured in kilometers (km). Then the function y = f(x) simply relates time flown to distance traveled.

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Question: how do you interpret the derivative of f?
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Interpretations come from looking at the units of measurement.
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What are the units of the derivative?
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The derivative is a ratio; change in y over change in x.
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If y is measured in km, then a change in y must be measured in km too.
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If x is measured in hours then a change in x must be measured in hours.
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So the derivative has units km/hour, that is kilometers per hour.
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So the derivative is simply speed or velocity - how fast you are going.
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Challenge: what's the derivative of velocity with respect to time?


next up previous
Next: 4.2 Up: 4. Previous: 4.
Richard Waterman
1999-06-01