2.2 Instantaneous Rate of Change - The Derivative
In the previous section, we saw how the average rate of change of a function is actually the slope of a line segment connecting two points on the function’s graph…
The slope of the green line segment is
This is also the average rate of change of f over the interval [x, x+h]. Next, we’ll see what happens when we take this to the limit (as h goes to 0). The result is a new function, derived from f, denoted f ‘ and called the derivative of f:
This is the instantaneous rate of change of f. As h becomes smaller, the average rate of change approaches the instantaneous rate of change. Geometrically, this is the slope of the tangent line to the graph of f at x.
In the following animation, you can see the slope of the secant line (shown in green) approach the slope of the tangent line (shown in red) to a function (shown in blue) at x=5, as h approaches 0:
The upper-case Greek letter delta, D, is often used to express the change in a variable. As x changes from x to x+h, the “change in x” is h, written Dx = h. We’ve seen that the corresponding “change in y”, Dy = f(x+ h) – f(x). This leads us to another very common notation for the derivative of a function, dy/dx (spoken “d y d x”).
Example 1: Compute the derivative of g(x) = 5x–2.
Example 2: Compute the derivative of F(x) = 3x2–2.
Example 3: Compute the derivative of
.
This one’s a good algebraic work-out, but you’ll soon learn a much easier and faster way to compute such derivatives (and I’m not referring to Maple, though that’s also of course easier and faster!). Sometimes I refer to this stage in the course as “stone-age calculus”. From crude stone implements, all later technology arises. When more sophisticated technology is available, older technologies fall into disuse. You’ll soon be calculating derivatives using much more sophisticated tools, which will make the calculations MUCH easier. But see if you can follow all the algebra in this derivation! ;-)
Later we’ll revisit this derivation, but it will only take a couple of easy steps!
2.2 Exercises Do all of them! :-)
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