So the function f (x) has a local maximum at (1,2), a point of inflection at (2,1.2), and a local
minimum at (3,0.4) where we have rounded to one decimal place when necessary.
In Figure N6–11b, the graph of f is shown again, but now it incorporates the information just
obtained using the FTC.
EXAMPLE 29
Readings from a car’s speedometer at 10-minute intervals during a 1-hour period are given in the
table; t = time in minutes, v = speed in miles per hour:
t 0 102030405060
v 26405510603245(a) Draw a graph that could represent the car’s speed during the hour.
(b) Approximate the distance traveled, using L(6), R(6), and T(6).
(c) Draw a graph that could represent the distance traveled during the hour.
SOLUTIONS:
(a) Any number of curves will do. The graph has only to pass through the points given in the
table of speeds, as does the graph in Figure N6–12a.
FIGURE N6–12a
(b) L(6) = (26 + 40 + 55 + 10 + 60 + 32) ·
R(6) = (40 + 55 + 10 + 60 + 32 + 45) ·
(26 + 2 · 40 + 2 · 55 + 2 · 10 + 2 · 60 + 2 · 32 + 45) =
(c) To calculate the distance traveled during the hour, we use the methods demonstrated in
Example 28. (We know that, since v(t) > 0, is the distance covered from time a to
time b, where v(t) is the speed or velocity). Thus,