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

176 Chapter 2 • The Derivative and Its Properties

PAGE s (miles)
172 31. Approximating Velocity An object moves on a line according (4, 2.7) (7, 2.7)
2
to the position function s(t) = 10t (s in centimeters and t in Princeton 2.7
seconds). Approximate the velocity of the object at time t 0 = 3 s Junction
RR station
by letting 1t first equal 0.1 s, then 0.01 s, and finally 0.001 s.
What limit does the velocity appear to be approaching? Organize
the results in a table.
1 (10, 1) (11, 1)
32. Approximating Velocity An object moves on a line according
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2
to the position function s(t) = 5 − t (s in centimeters and t in (0, 0) (13, 0)
Princeton 0
seconds). Approximate the velocity of the object at time t 0 = 1 by 0 4 7 10 13 17 t (minutes)
University
letting 1t first equal 0.1, then 0.01, and finally 0.001. What limit
does the velocity appear to be approaching? Organize the results
38. Jen walks to the deli, which is six blocks east of her house. After
in a table.
PAGE walking two blocks, she realizes she left her phone on her desk, so
173 33. Motion on a Line As an object moves on a line, its signed she runs home. After getting the phone, and closing and locking
distance s (in meters) from the origin after t seconds is given by
the door, Jen starts on her way again. At the deli, she waits in line
2
the position function s = f (t) = 3t + 4t. Find the velocity v to buy a bottle of vitaminwater TM , and then she jogs home. The
at t 0 = 0. At t 0 = 2. At any time t 0 . graph below represents Jen’s journey. The time t is in minutes,
34. Motion on a Line As an object moves on a line, its signed and s is Jen’s distance, in blocks, from home.
distance s (in meters) from the origin after t seconds is given by
(a) At what times is she headed toward the deli?
3
the position function s = f (t) = 2t + 4. Find the velocity v at (b) At what times is she headed home?
t 0 = 0. At t 0 = 3. At any time t 0 .
(c) When is the graph horizontal? What does this indicate?
35. Motion on a Line As an object moves on a line, its signed (d) Find Jen’s average velocity from home until she starts back to
distance s from the origin at time t is given by the position
get her phone.
1
2
function s = s(t) = 3t − , where s is in centimeters and t is in (e) Find Jen’s average velocity from home to the deli after
t getting her phone.
seconds. Find the velocity v of the object at t 0 = 1 and t 0 = 4.
(f) Find her average velocity from the deli to home.
36. Motion on a Line As an object moves on a line, its signed
s (blocks)
distance s from the origin at time t is given by the position
Deli 6 (16, 6) (19, 6)
√
function s = s(t) = 2 t, where s is in centimeters and t is in
seconds. Find the velocity v of the object at t 0 = 1 and t 0 = 4.
37. The Princeton Dinky is the shortest rail line in the country.
It runs for 2.7 miles, connecting Princeton University to the
(5, 2)
Princeton Junction railroad station. The Dinky starts from the 2
university and moves north toward Princeton Junction. Its
distance from Princeton is shown in the graph (top, right), where (7, 0) (25, 0)
House 0
the time t is in minutes and the distance s of the Dinky 0 5 10 15 20 25 t (minutes)
from Princeton University is in miles. (0, 0) (6, 0)
Applications and Extensions
39. Slope of a Tangent Line An equation of the tangent line to the
graph of a function f at (2, 8) is y = −5x + 12. What is f (2)?
′
40. Slope of a Tangent Line An equation of the tangent line of a
4
function f at (3, 4) is y = x + 1. What is f (3)?
′
3
41. Tangent Line Does the tangent line to the graph of y = x 2
MancusoMichael/AP Images 170 43. Respiration Rate A human being’s respiration rate R 3
at (1, 1) pass through the point (2, 5)?
Does the tangent line to the graph of y = x
42. Tangent Line
at (1, 1) pass through the point (2, 5)?
PAGE
(in breaths per minute) is given by R = R (p) = 10.35 + 0.59p,
where p is the partial pressure of carbon dioxide in the lungs.
Find the rate of change in respiration when p = 50.
(a) When is the Dinky headed toward Princeton University?
(b) When is it headed toward Princeton Junction? 44. Instantaneous Rate of Change The
volume V of the right circular cylinder of
(c) When is the Dinky stopped?
height 5 m and radius r m shown in the
(d) Find its average velocity on a trip from Princeton to Princeton 2
figure is V = V (r) = 5πr . Find the
Junction. 5 m
instantaneous rate of change of the
(e) Find its average velocity for the round trip shown in the volume with respect to the radius
graph, that is, from t = 0 to t = 13. when r = 3 m. r
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