Section 2.1 Two Problems with One Theme
V
97
•
=
(Vi+5h—4 \/16+5h+4
h=O
h
16+5h+4
limi
•
16+5h—16
inn
‘°h(l6+5h+4)
5
= urn
5
=
8
h-Ol6+5h+4
We conclude that the instantaneous velocity after 3 seconds is
second.
Velocity or Speed
the time being. we will use the
I
I ins
velocity and speed inter—
Ii:iiiieahly. Later in this chapter. we
sill distinguish between these two
ds.
centimeter per
ii
Rates of Change Velocity is only one of many rates of change that will he
important in this course: it is the rate of change of distance with respect to time.
Other rates of change that will interest us are density for a wire (the rate of
change of mass with respect to distance). marginal revenue (the rate of change of
revenue with respect to the number of items produced), and current (the rate
of change of electrical charge with respect to time). These rates and many more
are discussed in the problem set. in each case, we must distinguish between an
average rate ol change on an interval and an instantaneous rate of change at a
point.The phrase rate of change without an adjective will mean instantaneous rate
of change.
( oncepts Review
I. The line that most closely approximates a curve near the
-ilil P is
the
through that point,
2. More precisely, the tangent line to a curve at P is the limposition of the
line through P and Q as Q approaches
I’ thing the curve.
iiig
at
3. The slope rn
,, of the tangent line to the curve y
1
(c, f(c)) is given by rntan = )ifli
=
J’(v)
4. The instantaneous velocity of a point P (moving along a
line) at time c is the limit of the
on the time interval c to
c + h as h approaches zero.
Problem Set 2.1
hi
1
i
I ‘rob/ems I and 2, a tangent line to a c’iine is drawn. Estimate its
(slope = rise/run). Be carelid to tiote the difference in
ilcv on the two axes.
5.
6.
4
7. Consider y
draw the taligent ii,ie to the curie through the
Iii I’IUblefliS 3—6,
‘abated point and estimate its slope.
4.
3.
4
=
2
x
+
1.
(a)
(b)
(c)
(d)
Sketch its graph as carefully as you can.
Draw the tangent line at (1.2).
Estimate the slope of this tangent line.
Calculate the slope of the secant line through (1, 2) and
(1.01. (1.01)2 + 1.0).
(e) Find by the limit process (see Example I) the slope of the
tangent line at (1,2).
8. Consider = x
I.
(a) Sketch its graph as carefully
—
2-,
2
3
4
5
I’
-
as
you can.
(liapter 2 The I)erivative
—
9. Find the slopes of the tangent lines to the curve
1 at the points where x = -—2, —1, 0, 1, 2 (see Exam—
h) I )raw the tangent line at (2,7).
(c) Esi mate the slope of this tangent line.
(d) Calculate the slope ol the secant line through (2, 7) and
1.0).
(2.01, (2.01 )
(e) Find by the limit process (see Example 1) the slope of the
tangent line at (2,7).
=
-)K
v
—
10. Find the slopes of the tangent lines to the curve
3x it the points where x = —2, —1, 0, 1, 2.
=
)
12. Find the equation of the tangent line to y
I).
=
1/(x
—
1) at
II. Sketch the graph of y = 1/(x + 1) and then find the
(see Example 3).
equation of the tangent line at (i,
(0,
13. I xperin1ent suggests that a falling body will fall approx—
miately 6!’ eel in t seconds.
a) I-low far will it fall between I = 0 and 1 = 1’?
(e) Find its instantaneous velocity at
2 + Ii’?
2.003?
3’?
14. An object travels along a line so that its position s is
P + 1 meters after I seconds.
b ) I—I ow far will it fall between t = 1 and 1 = 2?
3?
t
(c) What is its average velocity on the interval 2
3.01?
t
(d) What is its average velocity on the interval 3
i’ = 3 (see Example 4).
=
t
(a) What is its average velocity on the interval 2
I
(b) What is its average velocity on the interval 2
(c) What is its average velocity on the interval 2 * t
(d) Find its instantaneous velocity at I = 2.
15. Suppose that an object moves along a coordinate line so
I hat its directed distance from the origin after t seconds is
\/2i + I feel.
(a) Fi id its instantaneous velocity at I = a, a > 0.
(Ii) When skill it reach a velocity of foot per second’? (see Ex
niiple 5.)
,2 + I grams after t hours.
17. A certain bacterial culture is growing so that it has a mass
16. If a particle moves along a coordinate line so that its
directed distance from the origin alter t seconds is (—P + 4t)
feet, when did the particle come to a momentary stop (i.e., when
did its instantaneous velocity become zero)’?
ol
2.01’?
t
(a) I-low much did it grow during the interval 2
(h) What was its average growth rate during the interval
2.01’?
t
2
(c) What was its instant:meous growth rate at I = 2’?
18. A business is prospering in such a way that its total (iccu
mulatcd) profit alter t years is 100012 dollars.
(a) I-low much did the business make during the third year (be
tween t = 2 and t = 3)’?
h) What was its average rate of profit during the first half of the
third year, belv.een I = 2 and I = 2.5’? (The rate will be in
dollars per 3 ear.)
(e) What was its instantaneous rate of profit at t = 2?
Mass is r g
xcm
19. A wire of length $ centimeters is such that the mass be
3
tween its left end and a point x centimeters to the right is x
grams (Figure 12).
E
Figure 12
(a) What is the average density of the middle 2-centimeter seg
ment of this wire? Note: Average density equals mass/length.
(b) What is the actual density at the point 3 centimeters from
the left end’?
—
20. Suppose that the revenue R(n) in dollars Irom producing
0.001w. Find the instan
n computers is given by R(n) = 0.4,i
taneous rates of change of revenue when n = 10 and n = 100.
(The instantaneous rate of change of revenue with respect to the
amount of product produced is called the marginal revenue.)
21. The rate of change of velocity with respect to time is
called acceleration. Suppose that the velocity at time t of a par
ticle is given by v(t) = 2(2. Find the instantaneous acceleration
when t = 1 second.
10;!
=
—
22. A city is hit by an Asian flu epidemic. Olficials estimate
that Idays alter the beginning of the epidemic the number of per
2t, when
sons sick with the flu is given by p(t) = 12012
t
40. At what rate is the flu spreading at time
0
20;i = 40’?
=
81)0
601)
401)
211))
4
16
Time in hours
12
2)1
24
23. The graph in Figure 13 shows the amount of water in a
city water tank during one day when no water was pumped into
the tank. What was the average rate of water usage during the
day? Flow fast was water being used at 8 AM.’?
C
H
Figure 13
0
20
II)
t
51)
(seconds)
41)
61)
70
90
‘40
24. Passengers board an elevator at the ground floor (i.e., the
zeroth floor) and take it to the seventh floor, which is 84 feet
above the ground floor. The elevator’s position s as a function of
time t (measured in seconds) is shown in Figure 14.
Figure 14
‘p
I
I
(
-
cit is the average velocity of the elevator from the time
In’ elevator began moving until the time that it reached the
v emit Ii floor’?
=
21)’?
(h) What was the elevator’s approximate velocity at time
(e) I lw many stops did the elevator make between the ground
Ituor and the seventh floor (excluding the ground and sev
nIh floors)? On which floors do you think the elevator
lapped?
-,
-‘
I
12)
Tr7’
T171
9)
----=----
II
L
3)
I
hI
—
—
—
211
241
-::i
151
Day oi the year
—
-
—
3_Il
-t
—
III)
—
—
27)
III
Missouri, as a function of time (measured in days begin
.‘s• Figure 15 shows the normal high temperature for St.
i(
41)
to
‘
iniuary 1).
‘Ins,
flnmi
J
lire 15
What is the approximate rate of change in the normal high
temperature on July 10 (i.e., on day number 191)’?
lml What is the approximate rate of change in the normal high
temperature on March 2 (i.e., on day number 61)? What are
he units of’ this rate of change?
I)
ti) In what months is there a moment when the rate of change is
equal to 0’?
1) In what months is the absolute value of the rate of change
t he greatest’?
-nt:Ige
Year
26. Figure 16 shows the population in millions of a developiii’, country for the years 1900 to 1999. What is the approximate
mile of change of the population in 1930? In 1990’? The percent
e row1h is often a more appropriate measure of population
p.m wth. This is the rate of growth divided by the population size
ii hat time. For this population, what was the approximate per
growth in 1930’? En 1990’?
(1
F igure 16
Section 2.1 Two Problems with One Theme
(a)
(I,)
99
27. I’igures 17a and 17b show the position s as a function of
time (for Iwo particles that are moving along a line. For each par
ticle, is the velocity increasing or decreasing’? Explain.
Figure 17
28. The rate of change of electric charge with respect to time
is called current. Suppose that
+ I coulombs of charge flow
through a wire in I seconds. Find the current in amperes
(coulombs per second) after 3 seconds. When will a 20-ampere
fuse in the line blow?
29. The radius of a circular oil spill is growing at a constant
rate of 2 kilometers per day. At what rate is the area of the spill
growing 3 days after it began’?
—
1
(h) 0
tangent
=
=
(c) 1
31. Draw the graph of y = 1(x)
line at
—
2 + 1. Then find
2x
(d) 3.2
(d) 4.2
sin x sin
2 2x. Then find
3
x
Use a graphing calculator or a CAS to do Problems 31—34.
30. The radius of a spherical balloon is increasing at the rate
ol 0.25 inch per second. If the radius is 0 at time t = 0, find the
rate of change in the volume at time I = 3.
(a)
the slope of the
32. Draw the graph of y = f(x)
the slope of the tangent line at
/3
(b) 2.8
(c)
(a)
33. If a point moves along a line so that its distance s (in feet)
from 0 is given by s = t + t cos
2 t at time I seconds, find its
instantaneous velocity at I = 3.
2. secant
34. If a point moves along a line so that its distance s (in me
ters) Irom 0 is given by s = (I + I )/(t + 2) at time t minutes,
find its instantaneous velocity at I = 1.6.
—
Answers to Concepts Review: 1. tangent line
3. [f(c + h)
f(c)j/h 4. average velocity