Please take this exam sometime before the quarter begins to determine
how prepared you are for this course. If you find you need work,
please review the appropriate high school material to the point of
understanding why you missed the questions you did. We will maintain
a steady pace during the course and it is very important that you do
not fall behind - the consequences may be harsh.
- What is 13/6?
- If a car holds a maximum of six people and 13 people wish to take
a trip together, how many cars will they need?
- How many times does 6 divide 13?
- What is the remainder of 13/6?
- Is 456 an integer?
- Is 456.34 an integer?
- Is 355/113 an integer?
- Is 456.34 a real number?
- A function is an unambiguous mapping of one or more values, taken
from a domain of values, to a range of output values. For example,
3*x+4 is a function that maps real values of x to
real values (the domain of x is all the real numbers, and the
range of 3*x+4 is all the real numbers). We will often use
f(x) to substitute for a complex function given input x
(for example, f(x) might be substituted for 3*x+4).
The following table expresses a mapping for some f(x)
(not 3*x+4):
| x |
1 |
2 |
3 |
4 |
5 |
| f(x) |
3 |
2 |
4 |
5 |
2 |
So, for example, a value of 3 for x maps to a value of 4 for
f(x). In other words, f(3)=4. The domain of
x is the integers from 1 to 5. The range of f(x) is
what?
- Can you modify the table in the previous question so that it does
not represent a function?
- Every function we will use has a name that uniquely identifies
it. We will want to use the value of one or more functions, given
particular arguments, when modeling complex systems. We will write
f(x) to express the fact that a function named f
maps one value. In that case x is said to be the argument of
f. Please realize that f(x) has no value until a
value for its argument x is set. However, in numerous
engineering applications, we are interested in visualizing the
relationship between x and f(x) (what are the values
of f(x) for a large collection of values of x). We
will write g(x,y) to express the fact that a function named
g has two arguments and so on.
For example, define a function called mod which maps two
integer arguments a and b to an integer whose value
is the remainder of a/b. Thus, the value of
mod(13,5) is 3. If the second argument is fixed (that is, a
constant), mod is a function of one argument. What is the
range of mod(x,5)?
- Define a function called floor which maps one real
number x to an integer which is the greatest that is less
than or equal to x. What is floor(2.34)?
- Define a function called ceil which maps one real number
x to an integer which is the smallest that is greater than or
equal to x. What is ceil(2.34)?
- A collection of objects is a set. We use curly braces
to denote a set. So, for example, {1,2,3} is a set
containing numbers 1, 2, and 3. Define the function union
that maps two sets to a set that contains all objects of both sets,
without duplicates. What is union({1,2,3},{2,3,4})?
- Define the function intersection to be a mapping from
two sets A and B to a set that contains all objects
that are members of both A and B. What is
intersection({1,2,3},{2,3,4})?
- Sometimes functions of two arguments are written with left
argument in front of and right argument behind a function symbol. For
example, instead of writing times(3,4), we write
3*4. In this case the function symbol is *.
Generally, a function symbol is referred to as an operator.
The arguments of an operator are referred to as operands.
Common operators are shown in the following table, with examples:
| Operator |
Meaning |
Example |
| + |
Numeric plus |
3+4 has value 7 |
| - |
Numeric minus |
4-3 has value 1 |
| / |
Numeric division |
8/5 has value 1.6 |
| * |
Numeric multiplication |
3*5 has value 15 |
| ^ |
Numeric exponentiation |
2^3 has value 8 |
| == |
Comparison |
2==1 has value false |
| > |
Comparison |
2>1 has value true |
| & |
Logical 'and' |
2==2 & 1==1 has value true |
| | |
Logical 'or' |
2==1 | 3==3 has value true |
Operators and their operands can be nested with parentheses specifying
the order in which the operations should be performed (inside out).
For example,
(2^(2+2)/(4+3)*5)/3 has value
0.15238...
Write a function, using operators from the table above, that maps a
year to a number expressing the world's population if population grows
at the rate of 1.3% per year and the population is known to be
6,500,000,000 in 2006.
- Using the function you developed in the previous question,
determine when the world's population was one eighth of, one quarter
of, one half of, and will be twice, four times, eight times what it is
in 2006. Display the results in a table.
- What simple conclusion can you make from the results of the table
you made in the previous question?
- Can you write a simple function that expresses that conclusion?
- Consider a right triangle with hypotenuse of length h
and sides of length a and b. What is the
relationship between a, b, and h?
- Write a function to determine the angle of the right triangle
that is opposite to the side of length a.
- What is log21024?
- What is 10! (ten factorial)?
- What is the probability that a flipped fair coin lands on heads?
- What is the probability that a roll of a pair of dice results in
a total of 7?