Example 4

Sketch the curve

The first two derivatives are

f'(x) = -2(x - 2)^-3

f"(x) = 6(x - 2)^-4.

The first and second derivatives are never zero, f(x) is undefined at x = 2. In our table we shall show the values of f(x) and its first two derivatives at a point on each side of x = 2. We shall also show the limits of f(x) and its first derivative as x → -∞ , x → 2^-, x → 2⁺, and x → ∞. (We will not need the limits of f"(x).)

The first line of the table, lim_x→-∞, shows that for large negative x the curve is close to 1 and its slope is nearly horizontal. The second line, x = 1, shows that the curve is increasing and concave upward in the interval (-∞, 2), and passes through the point (1, 2) with a slope of 2. The third line, lim_x→2-, shows that just before x = 2 the curve is far above the x-axis and its slope is nearly vertical. Going through the table in this way, we are able to sketch the curve as in Figure 5.3.2.

The curve approaches the dotted horizontal line y = 1 and the dotted vertical line x = 2. These lines are called asymptotes of the curve.