Page 22 - 2024-calc4e-SE proofs-4e.indd

Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

172 Chapter 2 • The Derivative and Its Properties

The average velocity of 96.0016 over the time interval [3, 3.0001], (1t = 0.0001 s)
should be very close to the instantaneous velocity of the rock at t = 3 s. As 1t gets
closer to 0, the average velocity gets closer to the instantaneous velocity. So, to obtain
the instantaneous velocity at t = 3 precisely, we use the limit of the average velocity as
1t approaches 0 or, equivalently, as t approaches 3.
2
2
1s f (t) − f (3) 16t − 16 · 3 2 16(t − 9)
lim = lim = lim = lim
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1t→0 1t t→3 t − 3 t→3 t − 3 t→3 t − 3
16(t − 3)(t + 3)
= lim = lim[16(t + 3)] = 96
t→3 t − 3 t→3
The rock’s instantaneous velocity at t = 3 s is 96 ft/s.
We generalize this result to obtain a definition for instantaneous velocity.

DEFINITION Instantaneous Velocity
If s = f (t) is the position function of an object at time t, the instantaneous
1s
velocity v of the object at time t 0 is defined as the limit of the average velocity
1t
as 1t approaches 0. That is,

1s f (t) − f (t 0 )
v = lim = lim
IN WORDS 1t→0 1t t→t 0 t − t 0
• Average velocity is measured over an
interval of time. provided the limit exists.
• Instantaneous velocity is measured at a
particular instant of time.
We usually shorten “instantaneous velocity” and just use the word “velocity.”
NOW WORK Problem 31.

EXAMPLE 5 Finding Velocity

Find the velocity v of the falling rock from Example 4 at:
(a) t 0 = 1 s after it begins to fall
(b) t 0 = 7.4 s, just before it hits the Colorado River
(c) At any time t 0 .

Solution
2
(a) Use the definition of instantaneous velocity with f (t) = 16t and t 0 = 1.
2
2
1s f (t) − f (1) 16t − 16 16(t − 1)
v = lim = lim = lim = lim
1t→0 1t t→1 t − 1 t→1 t − 1 t→1 t − 1
16(t − 1)(t + 1)
= lim = lim [16(t + 1)] = 32
t→1 t − 1 t→1
At 1 s, the velocity of the rock is 32 ft/s.
(b) For t 0 = 7.4 s,

2
1s f (t) − f (7.4) 16t − 16 · (7.4) 2
v = lim = lim = lim
1t→0 1t t→7.4 t − 7.4 t→7.4 t − 7.4
2
2
16[t − (7.4) ] 16(t − 7.4)(t + 7.4)
= lim = lim
t→7.4 t − 7.4 t→7.4 t − 7.4
= lim [16(t + 7.4)] = 16(14.8) = 236.8
t→7.4
NOTE Did you know?
236.8 ft/s is more than 161 mi/h! At 7.4 s, the velocity of the rock is 236.8 ft/s.
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