Download Contents - Wheaton College

Quantitative
Competency Handbook
Edition: 2000b
James E. Mann, Jr.
Wheaton College
Perfect behavior may be as unattainable as perfect gear changing
when we drive; but it is a necessary ideal prescribed for all men by
the very nature of the human machine . . . It would be dangerous to
think of oneself as a person of \high ideals" because one is trying to
tell no lies at all [instead of a few lies]. In reality you might just as
well expect to be congratulated whenever you do a sum, you try to
get it quite right. To be sure, perfect arithmetic is an \ideal"; you
will certainly make some mistakes in some calculations. But there
is nothing very ne about trying to be quite accurate at each step
in a sum. It would be idiotic not to try; for every mistake is going
to cause you trouble later on.
C. S. Lewis in Mere Christianity
Contents
1 Statistics
2
2 Simple Algebra
6
3 Calculator usage
8
4 Areas and Volumes
9
5 Ratio and Proportion
13
6 Exponential Decay and Growth
14
7
Spreadsheets
16
8
Counting
17
9
Dimensions and Units
19
10
Review of Functions and Calculus Concepts
22
Sample Test Questions
In the quantitative competency test, you will be asked questions from the
following areas: statistics, including elementary data displays, simple algebra,
calculator usage, areas and volumes, ratio and proportion, exponential growth
and decay, counting, and elementary spreadsheet usage. In the sequel, you will
nd a brief discussion of each of these areas. In addition, there will be references
to readily available books where you can nd more information.
1
Statistics
Numerical data is the subject of statistics. The idea is to take the data and
wring from it answers to certain questions. The most basic analysis of data
is its graphical display. There are three ways of plotting data which are commonly encountered: the frequency plot, the stem and leaf plot, and the box and
whisker plot. We will give an example of each of these for the following data:
97.6
99.2
98.6
99.0
98.2
98.7
98.4
98.5
100.7
98.0
101.0
98.7
99.8
99.0
98.6
97.5
To get the proper visual eect for the frequency plot and the stem and leaf
plot, we mark our number line o in coarse enough subdivisions so that the
data will \pile up" near values which occur most often. You can see the eect
in Fig.1. When making a frequency plot, you round the data to the nearest
tic mark on your number line. In Fig. 1, we have chosen divisions of 0.5 and
marked them on a horizontal number line. Then, we have represented each data
value with a point which is rounded to the nearest tic. The eect which you
see is a column of dots which pile the highest where the data value that occurs
most frequently lies.
97
98
99
100
101
Figure 1: A frequency plot for the data above.
The idea is to get the dots to pile high. Had we made ner subdivisions,
each dot might have its own tic mark, and there would have been no piling. In
the frequency plot, each value is represented, although they may be somewhat
3
rounded. You can glance at the plot and gather how many points are in the
data set and what value or values occur most frequently.
The stem and leaf plot is similar to the frequency plot in many ways. However, instead of making dots to represent each data point, we actually write the
dierence between the data value and the stem. See Fig. 2. Above the stem
98.5, we see the value .2; from this we conclude that 98:5 + :2 = 98:7 appears in
the data set. On a well made stem and leaf plot, the leaves should be arranged
.1
.0
97
.4
.2
.0
.2
.2
.1
.1
.0
98
.2
.0
.0
.3
99
.2
100
.0
101
Figure 2: The Stem and Leaf Plot of the data above. Each data value is plotted
by nding the value of the stem which is nearest and less than the
data value and subtracting the stem value from the data value to
form the leaf.
in the order of increasing value as shown in Fig. 2. A nished plot is often
made by rst putting the leaves with their stems and then arranging in proper
order. With this kind of plot, all of the data is visible and the exact values can
be recovered if necessary. Neither the frequency plot nor the stem and leaf plot
is a data compression plot. This contrasts with the box and whisker plot which
will be discussed next.
Before making the box and whisker plot, we must describe the median of
a set of data values. To nd the median, rst arrange the values in increasing
order. We rearrange our data as shown:
97.5
98.4
98.7
99.2
97.6
98.5
98.7
99.8
98.0
98.6
99.0
100.7
98.2
98.6
99.0
101.0
Q1
M
Q3
The median is the value in the middle of the data set. In the case being presented, there are an even number of data values. Therefore, there is no number
which is the middle number. In such a case, we average the numbers on either
side of the middle giving M = (98:6 + 98:7)=2 = 98:65. The rst quartile, Q1;
is the median of the smallest half of the data set, and the third quartile, Q3;
is the median of the largest half of the data set. The quartiles divide the data
set into four parts. Exactly half of the data values lie between Q1 and Q3.
If the set contains an odd number of values, the median is actually a member
4
of the set and no averaging is required, e.g. for the set 4, 7, 8, the value 7
is the median. In the case of an even number of values, the median value is
not included in either the smaller or larger half of the set when determining the
quartile values. The quartile values for the data are Q1 = (98:2+98:4)=2 = 98:3
and Q3 = (99:0 + 99:2)=2 = 99:1.
The box and whisker plot displays only Q1; M; Q3 and the smallest and
largest data value. The plot is made by taking the number line and making a
short mark above the line at Q1; M; Q3. These values are connected to make
the box. The whiskers extend from each end to the largest and smallest data
values. See Fig. 3.
97
98
99
100
101
Figure 3: The Box and Whisker Plot, showing Q1; M; Q3, and the largest and
smallest data values.
The box and whisker plot is a data compression plot because you cannot
tell how many values are used to make the plot and you cannot recover any of
the original data by examining the plot. You can tell that 50% of the data lies
between Q1 and Q3, the median , and the extreme values. Many times this is
enough information from a data set. Suppose that you were given the box and
whisker plot for all of your grades in some course. Do you think that such a
plot would tell a better story of your performance than does the average grade?
The box and whisker plot is particularly valuable for comparing two data sets.
In addition to graphical representations of data, there are two calculated
values used to characterize data. The rst is the mean or average, and the
second is the standard deviation.
The average is the sum of all of the data values divided by the number of
values. The average for our set is 1581:5=16 = 98:84, where the numerator is
the sum of the values. The average is an estimate of the middle of the data set
(as is the median). For many data sets, the average and the median are quite
close in value. When they dier by a large amount, you may be sure that the
data has an unusual distribution.
The standard deviation is a measure of how spread out the data is (in a sense,
Q1 and Q3 measure the same thing). The formula is rather complicated, but by
focusing on the deviation from the average, you will nd it is easy to remember.
Let us give the formula in general terms rather than specic numbers. Suppose
we have n data values called x1 ; x2 ; x3 ; ; xn . Then the average is given by
x + x2 + + xn
X = 1
:
n
5
The standard deviation is then given by
SD =
X )2 + (x2
(x1
X )2 + + (xn
n 1
1=2
X )2
:
We will write out the example completely for the data set 2, 4, 6, 8.
2+4+6+8
X =
= 5:
4
SD =
=
(2
p
5)2 + (4
5)2 + (6
3
5)2 + (8
5)2
1=2
20=3
If n is at all large, the standard deviation is an awkward calculation to make
with a calculator. There are variations of the formula which are easier for a
calculator, but readers are encouraged to learn to use the automatic feature of
their own calculators to nd the standard deviation. To learn to use this feature,
study the manual that came with your calculator. All calculators require (a)
that you clear the statistical storage area (sometimes this is done by entering
the statistical mode), (b) that you enter each data value with a special key such
as M +, (c) that you press the x key for the average and the key for the
standard deviation. In passing, we note that in some variations of the standard
deviation formula, there is an n rather than an n 1 in the denominator of the
formula. For our competency requirement, we will count either formula correct.
With practice, you can be expert at calculating the standard deviation.
Nevertheless, the calculation itself gives little insight into the signicance of the
term. Looking at the formula reveals that the SD is a measure of how far the
data values lie from their average; the larger the standard deviation, the more
spread out the data is. However, statisticians usually attach much more meaning
to the SD. If the data values are normally distributed1 (and many times, they
are) then 68% of the data values will lie within one standard deviation on either
side of their mean, i.e. 34% will be on each side of the mean. See the Figure 4.
Moreover, 98% will lie within two standard deviations on either side of the mean
and almost 100% will lie within three standard deviations. If we measured the
height of 75 students and calculated the mean to be 67 inches and the standard
deviation to be 3 inches, then we would expect that about 68% of all students
would have heights between 64 inches and 70 inches (one standard deviation on
either side of the mean). We emphasize that 68% of all normally distributed
data falls within one standard deviation of the mean has the same meaning as
the statement that if you ip a coin a number of times, one half of the ips will
be heads. Both statements are probabilistic and are meant in an average sense.
1 Normally
distributed means that the data is drawn from the normal distribution, i.e. a
frequency plot of such data would have the familiar bell shape. Average of data samples are
usually normally distributed.
6
68% of data
34%
mean
mean + σ
Figure 4: The number line showing one and two standard deviations of either
side of the mean for normally distributed data.
To learn more about statistical calculations, you may read any elementary
statistics book and your calculator manual. The following books are easily
understood.
2
Your own calculator manual for standard deviations.
Mosteller, F., Rourke, R.E.D., Thomas, G.B., Probability with Statistical
Applications, 2nd ed. Addison-Wesley, 1970
Simple Algebra
Important aspects of algebra deal with the straight line and with solving simple
equations.
To solve equations, you must isolate the unknown on one side of the equation. This isolation is accomplished by adding or multiplying both sides of the
equation by the same quantity. If the two sides are equal at the beginning, they
will remain equal as long as you perform the same operation to both sides of
the equation. Thus if
5x 4 = 7x + 14;
we may add 4 to both sides, giving
5x = 7x + 18:
Then we may subtract 7x from both sides to obtain
2x = 18:
Finally, dividing both sides by 2, we obtain
x = 9:
This general method works for all equations. Keep in mind that you must do
the same operation to both sides of the equation at each step.
There are many uses for the line and its equation. The simplest form of the
equation for a line is
y = mx + b:
7
There are several forms of the equation, but they can all be reduced to the one
above by multiplication and addition. I suggest that most people would be nd
it easier to remember only the one form. In the form above, m is the slope of
the line. If the value of x is increased by 1, then m is added to the value of y .
Among the questions which can be asked about the line are: identify the slope
from a specic equation, nd the equation given the coordinates of two points
on the line, nd the equation of the line given the slope and the coordinates of
a single point on the line. All of these questions can be answered by using the
form of the equation given above.
(a) Given: the equation 3x 5y = 10: Find the slope
Solution: Put the equation in standard form.
3
y= x
5
2:
Then, the slope = 3=5, the coeÆcient of x.
(b) Given: the line passes through the points whose coordinates are
(2; 3); (4; 2). Find the equation.
Solution: Begin with y = mx + b and substitute the values for x; y .
3 = 2m + b
2 = 4m + b
(2; 3)
(4; 2)
First, solve for m by subtracting the two equations to eliminate b.
5 = 2m
5
2
Now return to the rst equation and substitute m = 5=2 and solve
for b.
5
b = 8 and y =
x + 8:
2
(Check the answer by substituting (2; 3); (4; 2) into the equation
to make sure they work.)
m=
(c) Given: m = 3, and that the line passes through (4; 7). Find the
equation.
Solution: Start with y = mx + b. Substitute for m and the coordinates of the point.
7 = 3(4) + b
Therefore, y = 3x
b = 5:
5.
For help in algebra, your own book is a good rst resource. Otherwise, one
of the following may be helpful.
8
3
Fisher, Robert C. and Ziebur, Allen D. Integrated algebra trigonometry
and analytic geometry, 4th ed., Prentice Hall, 1982.
Ayres, Frank, Schaum's Outline of Theory and problems of rst year college algebra.
Calculator usage
You should be familiar with the most commonly used keys on your own calculator. These include +; ; ; ; y x ; , Store, Recall, and (). Most of us use
the rst four routinely.
One cause for error in calculator usage is the habit of writing intermediate
results and re-entering that result. Because people often round these written
values, large errors can occur if two nearly equal numbers must be subtracted.
There is also the increased possibility of a copy error. The way to avoid such
errors is to use either the () or the Store, Recall keys. For example,
8:378
4:452
6:219
5:977
might be found by calculating the numerator, writing down the result, calculating the denominator, writing this result, re-entering the numerator, pressing ,
re-entering the denominator and pressing =. This is a highly ineÆcient method
and may produce the errors mentioned above. The following sequence of key
strokes produces the result without writing down anything; the sequence uses
the () keys.
(8:378 6:219) (4:452 5:977) =
The same result can be obtained by using the Store and Recall keys (these
keys may have a dierent name on your calculator). Many calculators have a
number of storage areas (called registers) for numbers. Read your manual and
practice doing calculations without re-entering results. You will be asked to
do a complicated calculation and nd the answer correct to a large number of
signicant digits. The best way to nd such answers is to avoid data re-entry.
The y x key is the exponential key; it is used to raise numbers to a power. To
evaluate 2:364:52, the following key strokes are required: 2:36 y x 4:52 =. The
result is 48.47996. One reason to use this key is to answer questions like how
much money will I have in the bank in 16 years if the interest rate is 5% per year
and I deposit $25. The amount is 25(1 + :05)16 = 25(1:05)16 = 54:57. A similar
question is how many fruit ies will be alive in three days if their population
doubles every 6 hours and there are 18 ies initially. Solution: 3 days = 72
hours = 72/6 = 12 generations. Number of ies = 18(2)12 = 73728: Questions
about the solution process for such problems as interest and population will be
explained later. At the moment, we are concerned only with the use of the y x
key. Some times you will want to evaluate an expression like 3:2 7:6 . In this
case, the operations are done the same way except 7:6 must be entered into
the calculator. This 7:6 does not indicate subtraction but rather a negative
9
number. To enter a negative number, enter the number as positive followed by
, (on my calculator the key is CHS, standing for CHange Sign). The proper
sequence of key strokes is
3:2 y x 7:2 =
You will get 0:00023063 for an answer.
The only reference that will help you to use your calculator is the manual
that came with it. If the ideas above are new to you, work some of the exercises
in the calculator manual. These manuals usually have solved examples.
4
Areas and Volumes
A frequent use of mathematics is for the calculation of areas and volumes.
This activity is called mensuration. In many, perhaps most, instances an exact
formula is not required because an estimate of the quantity is good enough.
Questions such as how much wallpaper or paint is required for a particular wall,
how many gallons of water does the jacuzzi hold, do not require exact answers,
but the estimation must be good enough to avoid unnecessary expense.
The simplest formula for area is the one for the area of a rectangle, see panel
(a) of Fig. 5. The formula is
height
b
2
height
height
base
(a)
b
base
1
(b)
(c)
Figure 5: (a) A Rectangle, (b) A Parallelogram, (c) A Trapezoid.
Area = b h = base height:
In most cases, the base and the height must be measured in the same units
(inches, feet, yards, meters, etc.) so that area is measured in units which are
the square of the length unit. Fabric is usually sold in square yards; the amount
of surface covered by paint is given in square feet. If the height and base are
not in the same units, then you must convert one or both to a common unit.
For example, a driveway might be 8 feet wide and 45 yards long. The width of
a sidewalk might be measured in inches while its length would be measured in
feet. If you had an oer to cover a sidewalk with brick, it would be sensible to
calculate how much was being charged per square foot of walkway. It would be
necessary to change the width of the walk to feet in order to use the formula
for the area of the long, skinny rectangle of a walk. The area of a walk which
is 30 inches wide and 25 yards long is 187.5 square feet. As a matter of fact,
this is the approximate area of the walk even if it is curved so long as the
10
length is measured along the curve. A gure whose area formula is similar
to the rectangle is parallelogram shown in panel (b) of Fig. 5 . The formula
for area is again baseheight, but you must notice that the height is measured
perpendicular to the base. Finally, there is the trapezoidal gure shown in panel
(c). A trapezoid has two bases (the parallel sides) of diering length. These
lengths are designated b1 and b2 in the gure. The area is the height times the
average length of the bases.
b + b2
:
Area of trapezoid = h 1
2
The formula for the area of a triangle is easily deduced from any of the
panels in Fig. 5; by looking at the dotted lines in Fig. 5, we see that area of the
triangle is one half of the area of a rectangle or parallelogram or is the same as
the area of a trapezoid that one base equal to zero. Any of these observations
leads to the formula
1
Area of triangle = bh:
2
The area of circles and more complicated gures can be found by approximating them with rectangles. Indeed, this exactly what is done in calculus. For
our purposes, relatively crude approximations will suÆce. Look at the circle
in Fig. 6. The circle is contained precisely in a square whose area is d2 . It
d
(a)
d2
d
1
(b)
Figure 6: (a) A circle inscribed in a square. (b) An ellipse inscribed in a
rectangle.
is obvious that the area of the circle is somewhat less than that of the square.
Estimate how much less by making a guess about the fraction of the area of the
square which covered by the circle. I would guess 3=4 is covered. Using this, we
have
3
Area of circle d2 :
4
Perhaps it is as easy to remember the exact formula for the area of the circle.
If you do remember d2 =4, you can can calculate that our approximate area
is about 4.5% too small. For the ellipse inscribed in the rectangle, we use the
same guess
3
Area of ellipse d1 d2 :
4
Most people do not know the exact formula for the area of the ellipse but the
formula is similar to the one for a circle. The area of an ellipse is d1 d2 =4;
11
and again the estimate is 4.5% too small. For a more irregular shape that has
no bounding rectangle, estimate the area by area by dividing the shape into
a number of approximating rectangles, trapezoids, or triangles. If the shape is
plotted to scale, you must take the scale factor into account in order to calculate
the area properly. We will have more to say about these factors when we talk
about Ratio and Proportion later.
Perimeter of areas can be estimated by approximating the boundary by a
sequence of straight lines and then either measuring the lines or calculating
their lengths by using the theorem of Pythagoras for the relation of the sides
of a right triangle. If a; b; c are the sides and hypotenuse of a right triangle
respectively, Pythagoras stated that the quantities are related by
a2 + b2 = c2 :
In addition to using this formula to nd the lengths, it is frequently used by
carpenters to nd a right angle, using the numbers 3, 4, 5 for the sides of the
triangle. Since these numbers are related by the formula above the associated
triangle must be a right triangle.
The funny thing about the English system of units is that it may be relatively
easy to nd the area of certain shapes and rather diÆcult to express the area in
the units which are commonly used. An example is the acre. A typical city lot is
about 100 feet by 150 feet, or 15,000 square feet. But lots are usually discussed
in terms of acres. The problem is that the size of the acre is not related the to
square of the linear measure in a simple way. The original denition of an acre
was the amount of land which could be plowed by a yoke of oxen in a day2 . This
came to be standardized as piece of land measuring 4 poles by 40 poles. A pole
is 16.5 feet. Using this measurement, you can calculate that there are 43,560
square feet in an acre. There are various ways of remembering or calculating
the number of square feet in a acre. I remember that there are 640 acres in a
square mile and that a mile is 5280 feet in length. From there, one can easily
compute that an acre is 43,560 square feet. Find the size of a square which
contains an acre.
Volumes of simple shapes are also important for quantitative competence.
We will stick to simple box-like gures as shown in Fig. 7. Using the information
from the gure, we have
V = abc = length width height:
Many things have the basic shape of the box, e.g. an excavation for a basement,
concrete in a driveway or sidewalk; in these cases, you must again be careful
that all of the dimensions are given in the same units, but then the calculation
is straight forward. The common unit for concrete and for dirt (to be dug for a
basement) is the cubic yard.
A more complicated volume is the one mathematicians call a cylinder, illustrated in Fig. 7(b). The base of a cylinder may be any shape whatsoever,
2 Oxford
English Dictionary
12
a
height
h
width
b
s
base
d
area of base
(a)
(b)
(c)
Figure 7: (a) A box. (b) A general cylinder. (c) A box shape with the bottom
not parallel to the top.
but the sides must be composed of lines which are parallel to one another. The
formula for the volume is
V = area of base height:
If the one of the bases is not perpendicular to the sides, as in case of swimming
pool with a sloping bottom, the average height should be used in place of the
height. This average height might have to be estimated or guessed. In the case
of the swimming pool shown in Fig. 7(c), the volume is given by
V = ab(s + d)=2:
Similar to area, many things are sold in units of the cube of the length.
However, some things which are really sold by volume are measured in strange
units like uid ounces, gallons, or bushels. It is relatively easy to measure the
sides and depth of a swimming pool and calculate its volume in cubic feet,
but unless you know a conversion factor it is baing to express the volume in
gallons. If you know that a gallon of paint coats 400 square feet, it is not simple
to calculate the thickness of the coating without knowing the number of cubic
inches in the gallon. In the metric system, the basic unit of volume is the liter,
which is 1000 cm3 ; knowing that a liter of paint will coat 8 meter2 , leads to a
simple calculation of the coating thickness. The story of the denition of the
strange English units and there relation to the linear dimensions is sometimes
interesting, like the acre, and sometimes frustrating like the gallon. In fact,
the origin of the gallon appears to be unknown3 . \The term derives in English
from Old French/medieval Latin, and is thought by most authorities to refer
to some unit borrowed from Celtic sources { that is, the Gallic/Gaelic unit4 .
Some sources relate the word to gall like the bulge seen on some trees. What is
certain is that the unit is very old. The size of the gallon is sometimes related
to the weight of water; a gallon of water in the United States weighs about 8
pounds (this is a handy number to remember), but a gallon in Canada is larger
3 see Oxford English Dictionary.
4 Dent Dictionary of Measurement,
J.M. Dent, London 1994
13
than a U.S. gallon. Perhaps these anomalies exist because of the various kinds
of weights used over the centuries. At any rate, the denition of a gallon is now
231 cubic inches and the only way to get that number is to remember it or look
it up. The number is very close to the volume of a cylinder whose diameter is 7
inches and whose height is 6 inches (in fact, if you take = 22=7, the calculated
volume of the cylinder is exactly 231 cubic inches. Cylinders of that size would
have been easy to make, as would cylinders with half the diameter to give the
quart. Nevertheless, conversions would have been simpler had the gallon been
216 cubic inches (a cube 6 inches on the side) because 8 of them would have
made one cubic foot. To summarize, if you want to know how much hot water
is necessary for your jacuzzi, you will have to compute the volume in cubic feet
or cubic inches and use the fact that one gallon is 231 cubic inches to nd out
the number of gallons.
A rather challenging and practical problem is to answer a question like this:
is it cheaper to buy a gallon of gasoline in Canada for $1.85 or in the U.S. for
$1.35? A Canadian gallon (the Imperial gallon) is 277.25 cubic inches and a
Canadian dollar is worth only a fraction of a U.S. dollar, say 87 cents. You can
gure it out.
You can also calculate how many gallons of paint are required to cover an
acre if the coating is to be 0.004 inch thick.
5
Ratio and Proportion
The idea of ratio and proportion is extremely useful for all kinds of every day
calculations. Such calculations are made to reduce the size of a recipe, to change
the size of a dress pattern, to scale maps, and in many more situations. We will
illustrate with some examples:
1. A board 12 feet long weighs 15 lbs. How much will a board of the same
size by 16 feet long weigh?
Solution: Let x be the weight of the 16 ft board. Then
x
16
= :
12 15
Solving for x, we nd x = 20 lbs.
2. If the distance from Wheaton to Chicago is 40 km and 16 km is 10 miles,
how far is it from Wheaton to Chicago in miles?
Solution: Let x be the distance in miles.
40
x
=
16 10
Solving for x, we get x = 25 miles. You can reduce mistakes by noting
that on one side of the equality we have kilometers over kilometers and
on the other side miles over miles, making each side dimensionless.
14
3. If it costs 2 marks to drive 40 km, nd the cost per mile to drive. Assume
1 mark = $1.50.
Solution: From above, we found 40 km = 25 miles. Also 2 marks = $3.00.
Therefore, the cost per mile is 3.00/25 = $0.12 per mile.
4. Standing at point A, you notice that the top of a ag pole and the top of
an 8 foot pole are in exact alignment. See Fig. 8 for complete dimensions.
Find the height of the ag pole. Solution: Set up the proportion as follows.
X
8
22
60
Figure 8: Flag Pole and 8 foot pole.
x
8
=
22 60
Solving, we nd x = 21:8 feet.
When doing conversions such as kilometers per minute to miles per hour,
convert each dimension separately and then divide to obtain the result. Convert
75 miles per hour to kilometers per minute.
Solution: From above, 16 km = 10 miles. This implies that 75 miles = 120 km.
Also 1 hour = 60 minutes. Therefore, 75 miles per hour = 120=60 = 2 km per
minute.
We note that if all of the linear dimensions of an surface are scaled by factor
of k , the area is scaled by a factor of k 2 . If all of the dimensions of a solid are
scaled by k , the volume of the solid is scaled by k 3 . These important relations
are illustrated in Fig. 9
6
Exponential Decay and Growth
What do the population of an insect colony and the money in your savings account have in common? Answer: both grow exponentially. A quantity grows
exponentially if in a xed period of time, the amount increases by a xed percentage. If you put $100 into a savings account that pays 5% per year, in one
year you will have 100 + :05(100) = 105. If you leave the $105 in the bank for
another year, you will have 105 + :05(105) = 110:25. These calculations can be
15
k2 A
A
kc
c
V
b
k3 V
kb
a
ka
Figure 9: (a) A rectangle whose sides are both multiplied by k . (b) A box whose
sides are multiplied by k . Notice how the area and the volume of
the new gures are multiplied by k 2 and k 3 respectively.
done more eÆciently by noting that 100 + :05(100) = 100(1:05). Therefore, at
the end of two years, the amount is
100(1:05)2:
you can see that if you waited n years, you would have
100(1:05)n:
Any quantity, A, which gets larger according to the law Ak x where k > 1 is said
to grow exponentially. If 0 < k < 1, the quantity A is said to decay exponentially
because k x gets smaller as x gets larger.
In the case of a savings account, the bank usually gives the interest rate for
the year. However, they may compound the interest more often, e.g. quarterly
or monthly. This is always an advantage for the customer. If your $100 was
deposited in a bank that compounded quarterly at the 5% per year rate, you
would have 100(1:0125)4 = 105:09 at the end of the year. Note that if the
yearly rate is 5%, the bank usually takes this to mean that the quarterly rate
is :05=4 = :0125, the 4 in the exponent is there because there are four quarters
in the year. In two years, you would have 100(1:0125)8 = 110:45: The eect
of more frequent compounding mounts the longer one leaves the money in the
account. Sometimes a bank that compounds quarterly will say that the nominal
yearly rate is 5% but the \eective" rate is 5.09% (note 5.09% is the amount that
would have to be paid if the compounding was yearly rather than quarterly). On
some credit cards, the interest is 1.5% per month. This means that if you have
an unpaid balance of $100 each month for a year, you will owe 100(1:015)12 =
119:56, giving an \Eective" rate of 19.56%.
The exponential law is also used in idea of radioactive decay. For instance, we
are told that the Perry Mastodon is 11,000 years old on the basis of radioactive
carbon dating. The idea is that the amount of radioactive carbon which is
present in the mastodon began to reduce at the time of its death. This carbon
reduces to 1/2 its amount every 5700 years. 5700 years is the \half-life" of
16
radioactive carbon. The law looks A = A0 (1=2)n where A0 is the amount
present in the living animal, n is the number of half-lives since the animal died,
and A is the amount of radio carbon present after n half lives. Since a chemist
can determine A, and A0 is xed, it is possible to determine n and then the age.
We will be concerned only with the idea that the law is exponential.
A = A0 (1=2)n
where A0 is the original amount and n is the number of half-lives.
Example: Suppose a watch has a radioactive dial and the material of the dial
has a half-life of 12 years. What fraction of the original material will be left
after 18 years. Solution: 18 years is 18=12 = 1:5 half lives. There will be
(1=2)1:5 = 0:3535 or about 35% of the material remains. Radioactive materials
have half-lives ranging from millenia to small fractions of a second.
Population problems are similar to either growth as in savings accounts or
radioactive decay depending upon whether or not more people are being born
than are dying. One sometimes hears a statement that our school system has
been growing at the rate of 7% per year. If the enrollment was 4,750 when
such growth began, say 6 years ago, then the enrollment is now 4750(1:07)6. It
is customary to evaluate the growth exactly this way rather than to bother to
truncate the enrollment to the nearest whole number each year.
You will nd information on exponential growth in your calculator manual
and your college algebra book.
7
Spreadsheets
A spreadsheet is a computer program which is used to manipulate numbers.
Nearly anything that can be done with numbers can be done with a spreadsheet
including graphing, calculating standard deviations, and many many other operations. Most of the operations can be done with one or two mouse clicks or
commands. A spreadsheet might be called a \number processor" to put it on
an equal footing with a word processor. Like word processors, each spreadsheet
has its own set of commands but all spreadsheets are similar. At the moment,
we do not have the capability to test you on a variety of spreadsheet operations,
but we will explain what a spreadsheet is and how the formulas are written.
My conceptual view of a spreadsheet is a large sheet of graph paper. Each
square, called a cell, is addressed by telling in which column and in which row
the cell lies. The columns are named A; B; C; : : : and the rows are designated
by 1, 2, 3, . . . . Thus, B 3 refers to the cell that is in the second column and
the third row. Into the cells, we may put text, numbers, or formulas which
refer to contents of other cells. The text entries serve to identify the columns
for the reader and will not concern us. If the formula B 3 + A2 is found in
cell C 7 then when numbers are entered into cells A2 and B 3, their sum will
immediately appear in cell C 7. If we entered the numbers given in the statistics
section in cells A1 through A16, we could compute the average by writing the
formula @sum(A1::A16)/16 in some other cell, say A17. We could then make
17
1
2
3
..
.
A
97.6
98.2
100.7
..
.
B
A1
A2
A3
..
.
A17
A17
A17
C
..
.
Figure 10: A spreadsheet showing several cells; column A contains numbers
while column B contains formulas.
another column of numbers that contained the square of the dierence between
the numbers in A and their average. We could write in cell B 1 the formula
(A1 A17)2 , in B 2 the formula (A2 A17)2 and follow this pattern all the way
to row 16. You will notice that we must change the row number of the rst A,
i.e. A1, A2, etc. but A17 is the same in each row of the column B . When all
of this is complete, in B 17, you write @sum(B 1::B 16)/15 and you have most of
the standard deviation calculated; you still have to nd the square root. If you
did all of this and then wanted to change the numbers in column A you could
do so without changing anything else as long as you have any data set with 16
numbers. This sounds like a lot of work, but it is made much easier by the fact
that the formulas in column B need be written only once and they can then be
copied with the powerful copy commands in the spreadsheet; these commands
will change the row number automatically. Perhaps you have been wondering
about @sum(A1::A16). This is a command which tells the spreadsheet to Add
the numbers in cells A1 to cell A16; the A1::A16 gives the range of the numbers.
We can also make graphs using one column as x and another as y . Spreadsheets
commonly make line graphs, bar graphs, and pie graphs. These graphs are now
easily incorporated into word processor documents. Such incorporation makes
it easier to discuss and describe quantitative information. An appropriate graph
is often the best way to describe such data. Roughly, the procedure is to nd
the graphing icon and click. You must also specify which column is to be used
as x and which is y . If you have access to a spreadsheet, you should try to learn
to make graphs with it.
8
Counting
Counting seems like an especially simple task|1, 2, 3, . . . , what could be easier?
However, when the counting task exceeds the simple enumeration processes, it
can become among the most challenging in mathematics. A question like how
many distinct license plates can be produced using exactly three letters followed
by three digits is hard to do by simple counting because we would have to list
all of the plates rst and then do the counting. The list of plates would be a
very long one. However, a problem like this can help us distill principles that
will help us in thinking about more diÆcult problems.
18
The two principles that we will use are the multiplication principle and the
addition principle. The multiplication principle is stated: If an operation can
be performed in n ways, and after it is performed, a second operation can be
performed in m ways, then the two operations can be performed together in
n m ways. This principle can be generalized to any number of operations.
Applying the principle to the license plate problem above, we see six operations:
choosing a rst, second, third letter, and rst, second, third digit. Each letter
can be chosen in 26 ways (all letters are allowed for each choice) and each digit
can be chosen in 10 ways. Therefore, the number of ways a plate can be made
is
26 26 26 10 10 10 = 17; 576; 000:
The addition principle is stated: If two operations are mutually exclusive (if
one is done, the other cannot be done) and the rst can be done in n ways and
the second in m ways, then one operation or the other can be done in n + m
ways. For example, you may go to an icecream store where you can get either
a two-dip or a three-dip cone. All cones are made from chocolate, vanilla or
strawberry. You may have two or three dips of the same avor if you wish. How
many kinds of cones are available. The answer is there are 3 3 two-dip cones
and 3 3 3 three-dip cones. If you choose a two-dipper, you cannot have a
three-dip cone. How many choices do you have? The answer is 9 + 27 = 36.
These two principles can be applied separately and together to solve many
challenging counting problems problems. Here are a couple of recurring types
of counting situations. First, suppose I have fteen books, all dierent, on my
shelf. Let us assume that the fteen authors each have a separate initial from the
beginning of the alphabet, A through O. How many ways can it select 5 books
from the shelf? So ACDEL is one selection and DECAL is another and so forth.
The total number is the number of ways that I can select the rst book times the
number of ways I can select the second book times . . . times the number of ways
I can select the fth book. That number is 15 14 13 12 11 = 360; 360.
Here, the order of the selection of the books has been considered important.
Now suppose that you are going to select ve books and mail them to your
friend. How many possible selections can be made? Now, the order in which
the books are selected makes no dierence. ACDEL is the same set of ve
books as DECAL or LECAD. The answer is the number of ways the books can
be select when order is important divided by the number of ways those selected
books can be arranged. The number is 360360=(5 4 3 2 1) = 3003: The
distinction between order being important or not is a crucial one and you will
need to think about. If three members are being selected to take a petition to
the principal of the school, then the order of selection is unimportant. If you
are selecting candidates for president, vice president, and secretary, then order
is important.
19
9
Dimensions and Units
To be quantitatively competent in our society, you need to be somewhat familiar
with the way things are measured and the units of measurement. While this
topic is familiar to many of you, for others it will be helpful to have a few basics
spelled out. In the USA, we use the English system of units: inches, feet, yards,
pounds, tons, seconds, hours, etc., while most of the rest of the world uses metric
units like centimeters, meters, kilometers, grams, kilograms, etc. We all use the
same units of time. You should be familiar with the English system and with
some of the equivalents in the metric system. If you do not know the English
equivalences below, you should study them carefully.
English
12 inches
3 feet
5280 feet
16 ounces
2000 pounds
English
1 foot
1 yard
1 mile
1 pound
1 ton
The following are some equivalences between English and metric units.
English
1 inch
1 mile
2.2 pounds
Metric
2.54 centimeters
1.6 kilometers
1 kilogram
With these equivalences, you can gure out units of volume and area. For
instances, you can nd out how many inches are in a mile or how many ounces
are in a gram. In English units, there are measures of area and volume that
are not readily related to the measures of distance. Such measures as acre and
gallon are not simply related to length measures. However, it is fairly easy to
remember that 640 acres is the equivalent of one square mile and that 7.5 gallons
are about 1 cubic foot.
Unit conversions can be handled eectively by noticing that if the units are
written with each equivalence, they can be divided, multiplied and cancelled in
exactly the same way as fractions. To nd how many centimeters are in a mile,
you could write
1 mile = 5280 ft
(1)
1 ft = 12 in
(2)
1 in = 2:54 cm
(3)
If we substitute equation (2) into equation (1), we obtain
1 mile = 5280 12 in = 63; 360 in:
(4)
Next substitute equation (3) into equation (4) to obtain
1 mile = 63; 360 2:54 cm = 160; 934:4 cm:
20
(5)
This method always works. The units may be treated as algebraic quantities.
Here is a last example. There are 640 acres in a square mile.How many square
feet are there in an acre?
640 acre = (1 mi) 2
1 mi =5280 ft
Therefore,
640 acre = (5280 ft) 2 = 27; 878; 400 ft2
and finally,
10
Review of Functions and Calculus Concepts
22
d
• Quotient Rule:
dx
• Exponential Rule:
µ
f (x)
g(x)
¶
=
g(x)f 0 (x) − f (x)g 0 (x)
[g(x)]2
d f (x)
e
= ef (x) · f 0 (x)
dx
The derivative is used to find maximum values and minimum values of a function. For any function
f , a point c in the domain where f 0 (c) = 0 or f 0 (c) is undefined is called a critical point of the function.
Recall that if a function f is continuous at a point x = a then its graph has no jumps or holes at x = a.
If c is a critical point of a continuous function f , then the First-Derivative Test or Second-Derivative
Test can be used to determine whether or not f has a maximum or minimum value at f .
First-Derivative Test: Suppose c is a critical point of a continuous function f . If f 0 changes sign
from positive to negative at c, then f has a local maximum at c. If f 0 changes sign from negative to
positive at c, then f has a local minimum at c.
Second-Derivative Test: If f 0 (c) = 0 and f 00 (c) < 0, then f has a local maximum at c. If
f 0 (c) = 0 and f 00 (c) > 0, then f has a local minimum at c.
If we are given a velocity function v(t) and want to find a position function s(t), then we need
to find a function whose derivative is v(t), that is, find an antiderivative. We say that F (x) is an
antiderivative of f (x) if F 0 (x) = f (x). For example, an antiderivative of 3x2 is x3 . Note that x3 + 1
and x3 + π are also antiderivatives of 3x2 . The most general antiderivative of 3x2 is x3 + C where
Z
3
C is any constant. The antiderivative of x is denoted by
3x2 dx, which is called the indefinite
integral of 3x2 . There are many rules for finding antiderivatives, but two that follow readily from the
differentiation rules given earlier are the following:
R
n+1
• Antidifferentiation Power Rule: for any choice of constant n, with n 6= −1, xn dx = xn+1 + C.
Z
1
• Antidifferentiation Exponential Rule: for any choice of constant r 6= 0, erx dx = erx + C.
r
Let f be a continuous function defined for a ≤ x ≤ b. Then the definite integral of f from a to b,
Z b
f (x) dx, gives the signed area bounded by x = a, x = b, y = f (x), and the x−axis, i.e., the
denoted
a
area above the x−axis minus the area below the x−axis. It is computed using the following theorem:
Fundamental Theorem of Integral Calculus: For a continuous function f on an interval [a, b],
¯b
¯
f (x) dx = F (x)¯ = F (b) − F (a), where F is any antiderivative of f .
a
Z 2a
15
1
24
x4 ¯¯2
. Below are some of the important applications of
− =
x3 dx =
For example,
¯ =
4
4
4
4 1
1
the definite integral.
Z
b
Z
b
[f (x) − g(x)] dx.
• The area between two continuous curves f (x) ≥ g(x) on [a, b] is
a
Z
b
f (x) dx gives
• If f is a probability density function for a continuous random variable X, then
a
the probability that the values of X are in [a, b].
y
f (x)
a
x
b
P (a ≤ X ≤ b)
Z
• Since
F 0 (x)
= f (x), the Fundamental Theorem can be written as
a
b
F 0 (x) dx = F (b) − F (a).
This says that the definite integral of a rate of change of a function over [a, b] gives the total
Z b
s0 (t) dt = s(b) − s(a), the definite integral of
change of the function over [a, b]. For example,
velocity is the total change in position.
a
Examples (Section 10)
and 3 < t < 4 since v(t) = 0 on those intervals.
Example 3 If f (5) = 1, f 0 (5) = 6, g(5) = −3, and g 0 (5) = 2, find (f g)0 (5).
Solution By the product rule, (f g)0 (x) = f 0 (x)g(x) + f (x)g 0 (x). So we have
(f g)0 (5) = f 0 (5)g(5) + f (5)g 0 (5)
= 6 · (−3) + 1 · 2
= −18 + 2
= −16
√
xg(x) where g(4) = 2 and g 0 (4) = −5, find f 0 (4).
√
Solution We first rewrite x as x1/2 so that we can use the Power Rule. Then
Example 4 If f (x) =
d 1/2
x g(x)
dx
d 1/2
x · g(x) + x1/2 · g 0 (x) (by the Product Rule)
=
dx
= 1/2 · x−1/2 · g(x) + x1/2 g 0 (x) (by the Power Rule)
f 0 (x) =
Thus f 0 (4) =
1
2
· 4−1/2 · g(4) + 41/2 · g 0 (4) =
1
2
·
1
2
· 2 + 2 · (−5) = − 19
2 .
Example 5 Suppose that a mathematical model for the population of a city t years from now is
P (t) = 28, 800t1/3 + 3, 000, 000. Find and interpret P (8), P 0 (8) and P 00 (8).
Solution: Using the rules for differentiation, we find
1
· 28, 800 · t−2/3
3
= 9600t−2/3 , and
P 0 (t) =
P 00 (t) = −6400t−5/3
So P (8) = 28, 800(8)1/3 + 3, 000, 000 = 3, 057, 600, which will be the population of the city 8 years from
now.
We find P 0 (8) = 9600(8)−2/3 = 2400. Since P 0 (8) is the instantaneous rate of change of P (t) with
respect to t when t = 8, in eight years the population of the city will be increasing at a rate of 2400
people per year.
P 00 (8) = −6400(8)−5/3 = −200. After 8 years the growth rate is decreasing by about 200 people
per year per year. In particular, since P 00 (8) < 0 and P 0 (8) > 0, after 8 years the population is growing
but at a decreasing rate.
Example 6 A rancher has 1000 feet of fence and wants to build a rectangular enclosure along a straight
river. The side bordering the river does not require a fence. Find the dimensions of the field that will
make the enclosure as large as possible.
RIVER
y
y
x
Solution: We want to maximize the area A of the field. Let variables stand for the length and width:
x = length (parallel to the river)
y = width
The problem becomes to maximize A = xy subject to the constraint x + y + y = 1000. We must express
the area in terms of one variable. We use x = 1000 − 2y and substitute this into the area equation.
This gives A = (1000 − 2y)y = 1000y − 2y 2 . Note that the domain of A is 0 < y < 500. In order to
find the maximum of A we need to find the critical points of A, i.e., the points where A0 is zero or
does not exist. Since A0 = 1000 − 4y we have must solve 1000 − 4y = 0 for y, and find y = 250. Since
A00 = −4 < 0, the Second-Derivative Test shows that the area is indeed maximized when y = 250.
Thus the dimensions for length are x = 1000 − 2(250) = 500 feet, and width y = 250 feet.
¶
Z µ
1
7
2
3x + 4x − 2 dx.
Example 7 Find
x
Solution
¶
Z
Z µ
¡ 7
¢
1
7
2
3x + 4x2 − x−2 dx
3x + 4x − 2 dx =
x
x3 x−1
x8
(by the antidifferentiation power rule)
= 3 +4 −
−1
3
8
3 8 4 3 1
x + x + +C
=
x
3
8
Example 8 An object is observed to have an initial velocity of 200 m/s and to be accelerating at a
rate of 60 m/s2 . Find a formula for the velocity v(t).
Solution Velocity is the antiderivative of acceleration. Since v 0 (t) = 60 we find v(t) = 60t + C for
some constant C. We also know that v(0) = 200 which implies that C = 200. Thus v(t) = 60t + 200.
Example 9 A sports car can accelerate from a standing start to a speed of v(t) = −0.09t2 + 6t feet
per second after t seconds for 0 ≤ t ≤ 40. Find the distance the car will travel in the first 10 seconds.
3
2
Solution Since s0 (t) = v(t), we antidifferentiate v(t) to find s(t). Thus s(t) = − 0.09
3 t + 3t + c
where c is the initial position which we will take to be 0. In the first 10 seconds the car will travel
s(10) = −0.03(10)3 + 3(10)2 = 270 feet.
Z
Example 10 Find
3
x2 dx.
2
Solution Since f (x) = x2 is continuous on [2, 3] we can use the Fundamental Theorem.
Z
2
3
19
27 8
1
1 ¯¯3 1
− =
x2 dx = x3 ¯ = 33 − 23 =
3
3
3
3
3 2 3
Example 11 Write down the definite integral needed to find the area between y1 = x2 − 3x + 7 and
y2 = 2x + 7.
Solution By setting x2 − 3x + 7 = 2x + 7 we have x2 − 5x = 0 or x(x − 5) = 0. The two curves intersect
Z 5
[(2x + 7) − (x2 − 3x + 7)] dx.
at x = 0 and x = 5. Since y1 < y2 for 0 ≤ x ≤ 5, the area is given by
0
Example 12 The population of a small town is increasing at the rate of 2400e0.02t where t is the number
of years since 1990 when the population was 30,000. What does this model predict the population will
be in 2025?
Solution Let P (t) be the population at year t, where t is the number of years since 1990. We are given
that P 0 (t) = 2400e0.02t and P (0) = 30, 000. The total change in population from t = 0 to t = 35 is
Z 35
2400 0.02t ¯¯35
0
e
P (t) dt =
given by
¯ = 120, 000(e0.02·35 − e0 ) ≈ 121, 650. Thus the population in 2025
0.02
0
0
would be 121,650+30,000=151,650.
Example 13 Let X be defined as the number of inches of snow in a town during the first week of
January in each year. Suppose that a meteorologist looks at past weather records and determines that
the probability density function for X is f (x) = 6x(1 − x) on [0, 1]. Find the probability that the
snowfall during the first week of January will be between 0 and 0.4 inches.
Solution
Z
0.4
6x(1 − x) dx
P (0 ≤ X ≤ 0.4) =
0
Z
0.4
6x − 6x2 dx
¯0.4
¯
= 3x2 − 2x3 ¯
=
0
0
= (0.48 − 0.128) − 0
= 0.352
Thus the probability that the snowfall will be between 0 and 0.4 inches during the first week of January
is about 35%.
SAMPLE TESTS
(a) If Sammy leaves a 15% tip, write a formula that expresses how much
Sammy will leave the waiter.
(b) Write a formula that expresses how much Sammy will pay the cashier
if the tax rate is 6.5%.
5. A loan of $100 is made at an interest rate of 1% per month. What amount
must be repaid at the end of the year if no intermediate payments are
made?
6. Fruit Flies multiply rapidly. Every three days their population increases
by 40% (this rate accounts for deaths). If one starts with 100 fruit ies,
how many will there be in 21 days?
7. An unethical merchant advertises that every item in the store will be sold
at 30% o of the marked price. However, just before the sale begins, the
merchant increases the price of each item by 25%. If an item was marked
at $100 before the markup and sale, how much would one have to pay for
it during the sale?
8. Determine a value for the variable x in each equation that makes the
equation into a true statement.
(a) 5(x
(b)
2) = x
1
3
=
x x 5
9. The spread of a certain u virus is growing linearly. If the number of cases
grew from 200 on January 10 to 300 on January 14, on what date will the
number reach 475?
10. A gallon contains 231 cubic inches. How many gallons are there in a cubic
foot?
11. A agpole casts a 40 foot long shadow. At the same time a yardstick,
held to the ground vertically, casts a shadow 5 feet long. How tall is the
agpole?
12. A square piece of land which is 1/8 mile on a side contains exactly 10
acres. How many square feet are there in an acre? (1 mile = 5280 feet)
13. A straight line passes through the points (2; 5) and (6; 7). Find the slope
of the line.
14. A teacher must choose two girls and two boys to represent the class. There
are six girls and seven boys in the class. How many possible ways can the
teacher make the choice?
22
15. Find the median of the following data set:
18
29
34
16
23
25
13
17
9
10
16. Suppose that you are planning to draw a box and whisker plot of the data
for the previous problem. Where will the whiskers end?
17. It has been discovered that the average height of college students is 67
inches and that the standard deviation is 3 inches. Assuming the heights
are normally distributed, what percentage of students do we expect to be
taller than 70 inches?
18. A room is 20 feet long and 17 feet wide. The ceiling is 8 feet high. On one
wall, there is sliding door which is 8 feet wide and 80 inches tall. The walls
are to be papered with paper that costs $8 for 32 square feet. The door
is to be covered with material that costs $28 per square yard. Assuming
there is a 10% wasteage on both the wall paper and the drapery material,
how much will the materials cost for decorating the room?
19. A baseball player can throw a baseball from center eld to home plate
with an average speed of 60 miles per hour. If the ball travels 300 feet,
how many seconds is the ball in the air?
20. Between 2 and 4 o'clock, 3.2 inches of snow falls. If the snow continues
until 8 o'clock, how much additional snow will fall?
Answers: 1. B1: 5/6, B2: 1/6, B3: 14, B5: 100; 2. 1661:070737; 3. (a) 25, (b)
13.42 or 12; 4. (a) 0.15M, (b) 1.065M; 5. $112.68; 6. 1054; 7. $87.50; 8. (a) 2.5,
(b) 15/2; 9. January 21; 10. 7.48; 11. 24 feet; 12. 43,560; 13. 1/2; 14. 315; 15.
17.5; 16. 9, 34; 17. 16%; 18. $330.65 (door is not painted); 19. 3.4 sec; 20. 6.4in
23
Sample Competency Test II
1. Compute the values that would be generated by the column B formulas
shown on the spreadsheet below:
1
2
3
A
2
3
4
B
A1+A2*A3
A2+B1/A1
A1+B1/B2
Answers: (B1)
(B2)
(B3)
2. The following scores were made on a 50-point math test:
43 34 29 39
24 46 43 39
48 31 34 43
36 37 38 40
(a)
(b)
(c)
(d)
(e)
Find the mean score.
Find the median score.
Find the standard deviation of the scores.
Find the rst quartile of the scores.
Draw a box and whisker plot to represent the data set.
3. The formula to convert Celsius temperature to Fahrenheit is F = 9C=5 +
32. If the Fahrenheit temperature is 5, what is the Celsius temperature?
4. How much money must you put into into a savings account today in order
to have $5000 in ve years? Assume the annual interest rate is 5%, and
interest is compounded monthly. Ignore taxes.
5. A person deposits $1000 in the bank. Two years later, the person withdraws $200. How much money will the person have in the bank 6 years
after the original deposit? Assume the bank pays 6% interest, compounded
annually.
6. A pizza parlor oers a choice of eight dierent toppings, but allows a maximum of four toppings on any pizza. No double toppings are permitted,
and every pizza has cheese, which is not one of the toppings. How many
dierent types of pizzas could be ordered?
7. Using your calculator, nd the value of the following expression. Keep as
many decimal places as your calculator can produce.
1:4196:15=1:427
2:7
24
p
5:322:16= 97:79
8. Write an equation for the line that passes through the points (1; 2) and
( 3; 4).
9. A quart of milk occupies 57.75 cubic inches. How many gallons of milk
ll a cubic foot?
10. While on a 500 kilometer car trip in Canada, a driver uses 60 liters of
gasoline. How many miles per gallon did the car average? (Hint: 0.625
miles = 1 km, 0.264 gallons = 1 liter)
11. The length, width and height measurements in feet of a large box are 4,
3, 6. The outside surface including the top and bottom of the box are to
be painted. If a quart of paint covers 80 square feet, how many gallons of
paint should be purchased?
12. Air quality regulations in a certain town require that room capacities in
new buildings must be set low enough to allow 150 cubic feet of space for
each person in the room. If the room is 30 feet long, 40 feet wide, and 12
feet high, how many people will be allowed in the room?
13. A 10 foot ladder and a 16 foot ladder both lean against a building and
both form the same angle with the ground. The taller ladder reaches a
point 14 feet above the ground. What height on the building is reached
by the shorter ladder?
14. A person has scores of 51, 68, 77, and 75 on four tests. What score must
this person obtain on the fth test to achieve an average of 71?
15. A package courier charges $1.50 for the rst ounce and $0.60 for each
additional ounce or fraction thereof. The charge for a certain small package
is $5.10. How much did the package weigh?
16. A ight is scheduled from A to B. The normal time is 5 hours at a speed
of 500 miles per hour. After three hours, the pilot must slow to 400 miles
per hour due to weather. How late will the plane be?
17. The University of Chicago plans to increase its annual undergraduate costs
by 1.5%. The administration claims that the increase will amount to $480
additional charge for each student. What is the current annual cost for
each student?
18. A merchant adds 35% markup to the wholesale price. During a one day
sale, the merchant oers a 20% discount o of the marked retail price.
What will a customer pay for a shirt that originally cost the merchant $18
at wholesale?
19. The table below reects a sample of Wheaton College student opinion on
a certain issue.
25
Frosh/Soph Jr/Sr
Favor
35
62
Oppose
72
21
What percentage of students were Frosh/Soph?
20. If the set of test scores are normally distributed with a mean of 70 and
a standard deviation of 15, what is the probability of obtaining a score
higher than 55?
21. During a ve year period, a company had the following prots (in millions
of dollars). Make a line graph to display the results.
Year 1990 1991 1992 1993 1994
Prot 155 170 205 175 190
Answers: (1) B1: 14; B2: 10; B3: 3.4; (2)a: 37.75; b: 38.5; c: 6.36 or 6.16;
d: 34;
(3) 15; (4) $3896.03; (5) $1166.02; (6) 163; (7) 18.73215545 (8) y + 2 =
3(x 1)=2;
(9) 7.48; (10) 19.7; (11) 1 gal; (12) 96; (13) 8.75; (14) 84; (15) between 6 and 7
oz;
(16) 0.5 hours; (17) $32000; (18) $19.44; (19) 56% (20) 0.84
26
FUNCTIONS AND CALCULUS SAMPLE EXAM QUESTIONS
6. Find the critical points of f (x) = x3 + 3x2 − 24x.
(a) 0, 2, −4
(b) 2, −4
(c) 1, 2
(d) 2, 4, 8
√
−3 ± 105
(e) 0,
2
7. For the function f (x) which is graphed below, which of the following values is the greatest?
y
y = f (x)
x
−2
(a) 0
(b) f 0 (−2)
−1
1
(c) f 0 (1)
2
(d) f 0 (4)
3
4
(e) f (−2)
8. Suppose you are given the following information about f 0 (x). At which value(s) of x does f (x)
have a local maximum?
x<0
x=0
1<x<2
x=2
x>2
−
0
+
0
−
f 0 (x)
(a) x = 0
(b)x = 1
(c) x = 2
(d) x = 1 and x = 2
(e) f has no local maximum
9. A ball’s acceleration at time t is given by a(t) = 16t and its initial velocity is 35. Which of the
following functions gives the velocity v(t)?
(a) 35t2 + 16
(b) 16t2 + 35
Z b
10. Evaluate
9x8 dx.
(c) 8t2 + 35
(d) 8t2 + 35t + 35
0
(a)
72b7
(b) 9b9
(c) 91 b9
(d) b9
(e) −72b7
(e) 8t2 + 35t
11. A $10,000 car depreciates so that its value after t years is V (t) = 10, 000e−0.35t dollars. What is
the rate of change of its value after 2 years?
(a) decreasing by $1738 per year
(b) decreasing by $3500 per year
(c) decreasing by $4000 per year
(d) decreasing by $4966 per year
(e) decreasing by $9323 per year
12. An open top box with a square base x inches on each edge is to have a volume of 108 cubic
inches. In order to find the dimensions of the box that can be made using the smallest amount
of material, one would have to find the critical points of which of the following functions?
(a) x2 +
432
x
(b) 2x2 +
432
x2
(c) 2x2 +
108
x2
(d) x2 +
108
x
(e) x2 +
432
x2
13. Suppose oil is being extracted from a well at a rate of 0.1e0.5t millions of barrels per year. At
this rate how much oil will be extracted in the first five years?
(a) 1.2 million barrels
(b) 1.8 million barrels
(d) 2.4 million barrels
(e) 2.8 million barrels
(c) 2.2 million barrels
14. In a 5 year study, the number of years that a patient can expect to survive after an experimental
medical procedure is a random variable X with probability density function f (x) =
What is the probability that a patient survives between 2 and 4 years?
(a) 0.047
(b) 0.281
(c) 0.400
(d) 0.535
(e) 0.875
3 2
64 x
on [0, 5].
15. Which of the following definite integrals gives the area of the shaded region?
f (x)
y
−6 −5−4 −3 −2 −1
Z
g(x)
1
2
3 4
5 6
x
7
7
[f (x) − g(x)] dx
(a)
−6
Z
7
[g(x) − f (x)] dx
(b)
−6
Z
Z
−5
(c)
[f (x) − g(x)] dx +
Z
−5
Z
−5
−6
7
(e)
[f (x) − g(x)] dx
6
Z
6
[g(x) − f (x)] dx +
Z
7
[g(x) − f (x)] dx +
−6
(d)
Z
6
[f (x) − g(x)] dx +
−5
7
[g(x) − f (x)] dx
6
|f 0 (x) − g 0 (x)| dx
−6
4
Solutions to Sample Exam
1. (c)
10. (d)
2. (b)
11. (a)
3. (c)
12. (a)
4. (d)
13. (c)
5. (c)
6. (b)
14. (e)
7. (c)
15. (c)
8. (c)
9. (c)