Page 45 - How To Solve Word Problems In Calculus

(b) Since the position of the bicycle is known at every point
in time, the instantaneous velocity can be computed.
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Let s(t) = t . Then v (t) = s (t) = 2t

At 3 o’clock, t = 3so v (3) = s (3) = 6 mi/h.

The derivative determines the velocity of a moving object,
which indicates its speed as well as its direction. Often the
phrase “how fast” will be used in the formulation of a word
problem. “How fast” indicates that only the speed of the object
is desired. Unlike velocity, which may be positive, zero, or
negative, the speed of a moving object is always nonnegative.
Mathematically, speed is the absolute value of velocity:

speed =|velocity|

EXAMPLE 6
A particle’s position (in inches) along the x axis after t seconds
of travel is given by the equation

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x = 24t − t + 10
(a) What is the particle’s average velocity during the first
3 seconds of travel?
(b) Where is the particle and how fast is it moving after
3 seconds of travel?
(c) Where is the particle and how fast is it moving after
20 seconds of travel?
(d) When is the velocity of the particle 0? What is the particle’s
position at that instant?
(e) Describe the motion of the particle during the first
20 seconds of travel.

Solution
(a) The particle’s position when t = 0is x = 10 inches. When
t = 3, x = 199 inches.

x 199 − 10 189
v av = = = = 63 in/sec
t 3 − 0 3

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