Page 94 - Calculus Demystified

CHAPTER 3

Applications of

the Derivative

3.1 Graphing of Functions

We know that the value of the derivative of a function f at a point x represents the
slope of the tangent line to the graph of f at the point (x, f (x)). If that slope is
positive, then the tangent line rises as x increases from left to right, hence so does
the curve (we say that the function is increasing). If instead the slope of the tangent
line is negative, then the tangent line falls as x increases from left to right, hence
so does the curve (we say that the function is decreasing). We summarize:

On an interval where f > 0 the graph of f goes uphill.

On an interval where f < 0 the graph of f goes downhill.
See Fig. 3.1.
Withsomeadditionalthought,wecanalsogetusefulinformationfromthesecond

derivative. If f = (f ) > 0 at a point, then f is increasing. Hence the slope of the
tangent line is getting ever greater (the graph is concave up). The picture must be as
in Fig. 3.2(a) or 3.2(b). If instead f = (f ) < 0 at a point then f is decreasing.

Hence the slope of the tangent line is getting ever less (the graph is concave down).
The picture must be as in Fig. 3.3(a) or 3.3(b).
Using information about the first and second derivatives, we can render rather
accurate graphs of functions. We now illustrate with some examples.
EXAMPLE 3.1
2
Sketch the graph of f(x) = x .
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