Page 96 - Calculus Demystified

CHAPTER 3 Applications of the Derivative

First observe that f (x) = 2x. We see that f < 0 when x< 0 and f > 0 83

when x> 0. So the graph is decreasing on the negative real axis and the graph
is increasing on the positive real axis.

Next observe that f (x) = 2. Thus f > 0 at all points. Thus the graph is
concave up everywhere.
Finally note that the graph passes through the origin.
We summarize this information in the graph shown in Fig. 3.4.

Fig. 3.4

EXAMPLE 3.2
3
Sketch the graph of f(x) = x .
SOLUTION
2

First observe that f (x) = 3x . Thus f ≥ 0 everywhere. The function is
always increasing.
Second observe that f (x) = 6x. Thus f (x) < 0 when x< 0 and

f (x) > 0 when x> 0. So the graph is concave down on the negative real
axis and concave up on the positive real axis.
Finally note that the graph passes through the origin.
We summarize our findings in the graph shown in Fig. 3.5.
3
You Try It: Use calculus to aid you in sketching the graph of f(x) = x + x.
EXAMPLE 3.3
Sketch the graph of g(x) = x + sin x.
SOLUTION

We see that g (x) = 1 + cos x. Since −1 ≤ cos x ≤ 1, it follows that

g (x) ≥ 0. Hence the graph of g is always increasing.

Now g (x) =− sin x. This function is positive sometimes and negative
sometimes. In fact
− sin x is positive when kπ <x <(k + 1)π, k odd