463–474.3 As we can see by looking at the graph in Figure 4.2, over small enough intervals, the data points either lie on a line or lie close to some line that can be fitted to the data.
Trang 1change of −1 minute/day Certainly it is ridiculous to estimate that on November 6, 1998, the sun will set at 11:04 a.m (where 5:09 p.m −(1 minute/day)(365 days) = 5:09 p.m −
365 minutes = 5:09 p.m − (6 hours 5 minutes) = 11:04 a.m.)!
time
5:11
5:05
5:00
by days Dec.
Sunset time Greenwich Mean Time at 30° North Latitude time given to the nearest minute
Figure 4.2 Data from the 1998 World Almanac, pp 463–474.3
As we can see by looking at the graph in Figure 4.2, over small enough intervals, the data points either lie on a line or lie close to some line that can be fitted to the data However, the line that fits the data best varies with the interval chosen When looked at over the entire
interval from November 4 to December 25, the graph does not look linear.
In the last example we looked at a discrete phenomenon and made predictions based
on the assumption of a constant rate of change over a small interval In the next example we’ll look at a continuous model
EXAMPLE 4.2 Brian Younger is a high-caliber distance swimmer; in competition he swam approximately
1 mile, 36 laps of a 25-yard pool.4If he completes the first 24 laps in 12 minutes, what might you expect as his time for 36 laps?
SOLUTION Knowing that Brian is a distance swimmer, it is reasonable to assume that he does not
tire much in the last third Assuming a constant speed of 12 laps every 6 minutes (or 120 yards/minute) we might expect him to finish 36 laps in about 18 minutes We would feel less confident saying that he could swim 4 miles if given an hour and 12 minutes, or 8 miles
if given 2 hours and 24 minutes
A quantity that changes at a constant rate increases or decreases linearly.
If the rate of change of height with respect to time,
height
time
,
is constant over a certain time interval, then height is a linear function of time on that interval
3 These times might look suspect to you; the sun begins to set later by December 10, well before the winter solstice Do not
be alarmed: Sunrise gets later and later throughout December and continues this trend through the beginning of January.
4
Trang 2If the rate of change of position with respect to time,
position
time
,
is constant, then position is a linear function of time
Think back to the bottle calibration problem For a cylindrical beaker, the rate of change
of height with respect to volume is constant; height is a linear function of volume
height
Figure 4.3
Many of the functions that arise in everyday life (in fields like biology, environmental
science, physics, and economics) have the property of being locally linear What does this
mean?
“Local” means “nearby” and “linear” means “like a line.”5So a function is locally linear
if, in the immediate neighborhood of any particular point on the graph, the graph “looks like
a line.” This is not to say that the function is linear; we mean that near a particular point the
function can be well approximated by a line In other words, over a small enough interval, the rate of change of the function is approximately constant
Graphically this means that f is locally linear at a point A if, when the graph is sufficiently magnified around point A, the graph looks like a straight line.6The questions
of which line best approximates the function at a particular point and just what we mean by
“nearby” are very important ones, and we will examine them more closely in chapters to come First, let’s look back at Examples 4.1 and 4.2
In Example 4.1 the runner estimating the time of sunset is assuming local linearity; her assumption leads to a prediction that is only 1 minute off when she predicts just a few
days ahead The function is only locally linear; the idea of “locality” does not extend from
the first three readings in early November all the way to Christmas day In Example 4.2 the assumption that the swimmer’s pace is maintained for another 5 minutes is an assumption
of local linearity
You probably have made many predictions of your own based on the assumption of local linearity without explicitly thinking about it For instance, if you buy a gallon of milk and you have only a quarter of a gallon left after three days, you might figure that you’ll be out of milk in another day Here you’re assuming that you will consume milk at a constant rate of14 gallon/day Or, suppose you come down with a sore throat one evening and take three throat lozenges in four hours You might take six lozenges to work with you the next day, assuming that you’ll continue to use them at a rate of34 lozenge/hour for eight hours
5By “line” we mean straight line.
6
Trang 3Clearly you wouldn’t expect to pack six lozenges with you every day; you’re assuming only
locallinearity
Many examples of the use of local linearity arise in the fields of finance and economics Investors lay billions of dollars on the line when they use economic data to project into the future The question of exactly how far into the future one can, within reason, linearly project any economic function based on its current rate of change, and by how much this projection may be inaccurate, is a matter of intense discussion
Local linearity plays a key role in calculus The problem of finding the best linear approximation to a function at a given point is a problem at the heart of calculus In order
to work on this keystone problem, one must be very comfortable with linear functions So, before going on, let’s discuss them
P R O B L E M S F O R S E C T I O N 4 1
1 Lucia has decided to take up swimming She begins her self-designed swimming program by swimming 20 lengths of a 25-yard pool Every 4 days she adds 2 lengths
to her workout Model this situation using a continuous function In what way is this model not a completely accurate reflection of reality?
2 Cindy quit her job as a manager in Chicago’s corporate world, put on a backpack, and is now traveling around the globe Upon arrival in Cairo, she spent $34 the first day, including the cost of an Egyptian visa Over the course of the next four days, she spent a total of $72 on food, lodging, transportation, museum entry fees, and baksheesh (tips) She is going to the bank to change enough money to last for three more days in Cairo How much money might she estimate she’ll need? Upon what assumptions is this estimate based?
3 It is 10:30 a.m Over the past half hour six customers have walked into the corner delicatessen How many people might the owner expect to miss if he were to close the deli to run an errand for the next 15 minutes? Upon what assumption is this based? Suppose that between 9:30 a.m and 11:30 a.m he had 24 customers Is it reason-able to assume that between 11:30 a.m and 1:30 p.m he will have 24 more customers? Why or why not?
4.2 LINEAR FUNCTIONS
The defining characteristic of a linear function is its constant rate of change
EXAMPLE 4.3 For each situation described below, write a function modeling the situation What is the rate
of change of the function?
(a) A salesman gets a base salary of $250 per week plus an additional $10 commission for every item he sells Let S(x) be his weekly salary in dollars, where x is the number of items he sells during the week
Trang 4(b) A woman is traveling west on the Massachusetts Turnpike, maintaining a speed of 60 miles per hour for several hours Her odometer reads 4280 miles when she passes the Allston/Brighton exit Let D(t) be her odometer reading t hours later
SOLUTION (a)
salary = base salary + commission salary = base salary + dollars
item
(items) S(x) = 250 + 10x
The rate of change of S = S
x = $10 per item
(b) odometer reading = (initial odometer reading) + (additional distance traveled) odometer reading = (initial reading) + miles
hour
(hours) D(t ) = 4280 + 60t
The rate of change of D =Dt = 60 miles per hour
D e f i n i t i o n
f is a linear function of x if f can be written in the form f (x) = mx + b, where m
and b are constants
The graph of a linear function of one variable is a straight nonvertical line; conversely, any straight nonvertical line is the graph of a linear function As we will show, the line
y = mx + b has slope m and y-intercept b The slope corresponds to the rate of change
of y with respect to x Every point (x0, y0)that lies on the graph of the line satisfies the equation In other words, if (x0, y0)lies on the graph of y = mx + b, then y0= mx0+ b Conversely, every point whose coordinates satisfy the equation of the line lies on the graph
of the equation As discussed in Chapter 1, this is what it means to be the graph of a function The following are examples of equations of lines and their graphs
1 –1 1 2
f
x
Figure 4.4 f (x) = 2x − 1
m = 2, b = −1
Trang 5x
Figure 4.5 f (x) =√10
m = 0, b =√10
1 1
f
x
Figure 4.6 f (x) = −0.5x
m = −0.5, b = 0
The slope of a line is the ratio
rise run, or
change in dependent variable change in independent variable.
If y is a linear function of x, then the slope is
change in y change in x, or
y
x
We will now verify that if y = mx + b, then the constant m is the slope of the line
Verification: Suppose (x1, y1) and (x2, y2)are two distinct points on the graph of
y = mx + b We want to show that the rate of change of y with respect to x is m, regardless
of our choice of points
Since y = mx + b, the points (x1, y1)and (x2, y2)can be written as (x1, mx1+ b) and (x2, mx2+ b)
Trang 6∆ x
(x2, y2) = (x2, mx2 + b)
(x1, y1) = (x1, mx1 + b)
Figure 4.7
Therefore,
change in y change in x =y
x
=yx2− y1
2− x1
=(mx2+ b) − (mx1x + b)
2− x1
=mx2− mx1
x2− x1
=m(x2− x1)
x2− x1
= m
We have shown that the slope of the line y = mx + b is indeed m The slope of a line
is a fixed constant, regardless of how it is computed
To verify that b is the y-intercept of the line, we set x = 0 Then y = m · 0 + b = b
(Remember that the y-intercept is the value of the function on the y-axis, i.e., when x = 0.)
EXAMPLE 4.4 3x + 2y = 7 is the equation of a line Find the slope and the x- and y-intercepts
SOLUTION Put the equation into the form y = −32x + 72 We can then read off the y-intercept as
7
2and the slope as −32 Find the x-intercept by setting y = 0:
3x + 2(0) = 7 3x = 7
x = 7
3. Alternatively, begin with 3x + 2y = 7 and find the x-intercept by setting y = 0 and solving for x: (73, 0) Find the y-intercept by setting x = 0 and solving for y: (0,72) (The x- and y-intercepts can be useful in graphing the line.) Given any two points, you can find the slope by computing xy For instance, given the points (73, 0) and (0, 72), the slope is computed as follows
y
x =
7
2− 0
0 −73 =
7 2
−73 =
7 2
−3 7
= −3 2
Trang 7x
1
1 2
3 4 –1
–2
2 3
5
7 7
Figure 4.8
It is not necessary to use the x- and y-intercepts in order to calculate the slope; any two points will do
EXERCISE 4.1
is a linear equation In other words, verify that it can be written in the form y = mx + b If
ax + cy = d, what is yx?
Lines and Linear Equations
EXERCISE 4.2 Graph the following lines Use three sets of axes, one for each of the Parts I, II, and III
I a) y = x b) y = 2x c) y = 3x d) y = 0.5x e) y = −2x f) y = −0.5x
II a) y = 2x b) y = 2x + 1 c) y = 2x − 2 III a) y = 3 b) x = 4 c) x = 0 d) y = 0
Notice that the results of this exercise are consistent with the principles of stretching, shrinking, flipping, and shifting discussed in Section 3.4
From this exercise you can observe how the constant m corresponds to the steepness
of the line
m = 0 ⇒ The line is horizontal
m >0 ⇒ The line rises from left to right, so the function is increasing
The more positive m is, the steeper the rise of the line
m <0 ⇒ The line falls from left to right, so the function is decreasing
The more negative m is, the steeper the fall of the line
The closer m is to zero, the less steep the line
Vertical Lines
A vertical line is the graph of an equation of the form x = constant Notice that a vertical line is not the graph of a function of x One x-value is mapped to infinitely many y-values;
a vertical line certainly fails the vertical line test!
Trang 8x
slope undefined
x y=1
slope = 0
Figure 4.9
Slope =yx, so the slope of a vertical line is undefined, while the slope of a horizontal line
is zero
Parallel and Perpendicular Lines
If L1and L2are nonvertical lines with slopes m1and m2, respectively, then
L1is parallel to L2if and only if m1= m2, i.e., their slopes are equal
L1 is perpendicular to L2 if and only if m1= −1m 2, i.e., their slopes are negative reciprocals
Fundamental Observation
A line is completely determined by either
i the slope of the line and any one point on the line, or
ii any two points on the line
This geometric observation can be translated as follows If there is a linear relationship between two variables, then that relationship can be completely determined provided
i one data point is known and the rate of change of one variable with respect to the other
is known, or
ii two data points are known
Finding the Equation of a Line
i Suppose we know the slope of the line, m, and a point on the line, (x1, y1) We can find the equation of the line by using either of the following two methods
. Method 1:The equation of any nonvertical line can be put into the form y = mx + b
If we know the slope, we know m Our job is to find b Because (x1, y1)is a point on the line, (x1, y1)satisfies the equation y = mx + b Therefore, y1= mx1+ b, where
x1, y1, and m are constants We can solve for b, the only unknown, and then write
y = mx + b
. Method 2:The slope =yx Point (x1, y1)is a fixed point on the line Therefore, if (x, y) lies on the line,
m =y − y1
x − x1. This is written
y − y1= m(x − x1)
Trang 9to make it clear that the domain is all real numbers Solving for y will put this into the form y = mx + b
The equation y − y1= m(x − x1)is called the point-slope form of the equation
of a line
ii Suppose we know two points on a line: (x1, y1)and (x2, y2) We can find the slope of the line:
y
x =yx2− y1
2− x1= m.
Knowing the slope and a point (two points, actually), we can continue as described in part i
EXAMPLE 4.5 Find the equation of the line passing through points (2, −3) and (−4, 5)
SOLUTION Calculate the slope
m = −3 − 5
2 − (−4)=
−8
6 = −4
3, so
y = −4
3x + b
Use the fact that point (2, −3) lies on the line to find b
−3 = −4
3(2) + b ⇒ b = −3 +8
3= −9
3+8
3 = −1 3
y = −4
3x − 1
3. Alternatively, after finding m = −43we can use Method 2 above
−4
3 =y + 3
x − 2
y + 3 = −4
3(x − 2)
We leave it up to you to verify that the equations are equivalent
T h e E q u a t i o n o f a L i n e
Slope-intercept form of a line Point-slope form of a line
where m is the slope and b is the y-intercept where m is the slope and (x1, y1)is a point
on the line
Trang 10Scientists and social scientists often make general models in which constants or pa-rameters are represented by letters.7Cases in which the slope and/or points of a line are not given as numbers but as letters representing constants typically are stumbling blocks for students trying to determine the equation of the line Unless you think carefully and clearly,
it is easy to lose track of the main features of the equation:
the two variables, the constants that are given, and the unknown constants that you are trying to find
If you plan to consider slope, you must decide which variable will play the role of the inde-pendent variable and which will be treated as the deinde-pendent variable (If you’re accustomed
to x’s and y’s, then determine which variable will play the role of x and which will play the role of y.) By taking inventory, keeping your goal firmly in mind, and determining the steps you’ll take to reach that goal, you ought to be able to proceed fearlessly Try the following exercises
EXERCISE 4.3
i Find the equation of the line through the points (2π , π2)andπ,π1
ii Find the equation of the line through the point (a, c2)having slope p
iii Find the equation of the line through (m, n) with slope b
CAUTIONHere m and b are playing nonstandard roles To avoid confusion, try a strategic maneuver; use something such as y = Ax + C as your “reference” equation to avoid getting the m’s and b’s in the problem confused with the slope and the y-intercept.)
EXERCISE 4.4 Over the interval [0, T ] a horse’s velocity, v, is a linear function of time, t
(a) At time t = 0, the horse’s velocity is v0 At time t = T, his velocity is vT Find v(t) on the interval [0, T ] Note: v0, T, and vT are all constants
(b) What can you say about the horse’s acceleration?
Answers to Selected Exercises
Answers to Exercise 4.3
i y =ππ3−12 x − π2+π2
ii y = px + c2− pa iii y = bx + n − mb
Answers to Exercise 4.4:
(a) v(t) = mt + b because v is a linear function of t Our job is to find m and b We are given two points, (0, v0)and (T, vT) (Points are written in the form (time, velocity) because time is the independent variable.) In other words, t plays the role of x and v
7 Physicists use g to denote acceleration due to the force of gravity Biologists might denote the carrying capacity of an ecosytem for a certain species by a letter, such as C, without specifying the value of C Chemists use r for Avogadro’s number, the number