Calculus 2nd Edition Rogawski

SECTION 3.5 Higher Derivatives 141
(A) Large second derivative:
Tangent lines turn rapidly.
FIGURE 3
(B) Smaller second derivative:
Tangent lines turn slowly.
(C) Second derivative is zero:
Tangent line does not change.
EXAMPLE 6 Identify curves I and II in Figure 4(B) as the graphs of f ′ (x) or f ′′ (x)
for the function f (x) in Figure 4(A).
Solution The slopes of the tangent lines to the graph of f (x) are increasing on the interval
[a,b]. Therefore f ′ (x) is an increasing function and its graph must be II. Since f ′′ (x) is
the rate of change of f ′ (x), f ′′ (x) is positive and its graph must be I.
y
Slopes of tangent
lines increasing
y
I
II
x
a
b
a
b
x
(A) Graph of f (x)
(B) Graph of first two derivatives
FIGURE 4
3.5 SUMMARY
• The higher derivatives f ′ ,f ′′ ,f ′′′ ,...are defined by successive differentiation:
f ′′ (x) = d
dx f ′ (x),
f ′′′ (x) = d
dx f ′′ (x), . . .
The nth derivative is denoted f (n) (x).
• The second derivative plays an important role: It is the rate at which f ′ changes. Graphically,
f ′′ measures how fast the tangent lines change direction and thus measures the
“bending” of the graph.
• If s(t) is the position of an object at time t, then s ′ (t) is velocity and s ′′ (t) is acceleration.
3.5 EXERCISES
Preliminary Questions
1. On September 4, 2003, the Wall Street Journal printed the headline
“Stocks Go Higher, Though the Pace of Their Gains Slows.” Rephrase
this headline as a statement about the first and second time derivatives
of stock prices and sketch a possible graph.
2. True or false? The third derivative of position with respect to time
is zero for an object falling to earth under the influence of gravity.
Explain.
3. Which type of polynomial satisfies f ′′′ (x) = 0 for all x?
4. What is the sixth derivative of f (x) = x 6 ?