Lecture 45c Theory Questions
1. Consider two parabolas. The focus of the first is on the y-axis near the x-axis. The focus
of the second is on the y-axis and far from the x-axis. How do the shapes of these two
parabolas differ? (Hint: consider the values

Lecture 40 Theory Questions
1. True or False: A 2x2 matrix can be equal to a 3x3 matrix. Explain your reasoning.
2. True or False: A 2x2 matrix and a 3x3 matrix can be added together. Explain your
reasoning.
3. Let A be a 5x6 matrix with 32 = 5 and B be a

Lecture 33 Theory Questions
1) Assume that and 1 exist.
a) If (2) = 3 what is 1 (3) = ?
b) if () = what does 1 () = ?
c) if 2 = what is 2 () = ?
2) Suppose that point (2,3) is on the graph = . Find a point on the graph of
().
3) How are the asymptotes of

Lecture 38 Theory Questions
1. By definition, which of the following are matrices?
a. [1
2 3]
1
b. [2]
3
1
c. [
4
1
d. [
2 3
]
5 6
2 3
]
4
5
1 2 3
e. [ 4 5 ]
6
2. Suppose that a system of equations consists of four linear equations with five variables.
Wh

Lecture 45d
Hyperbola
Hyperbolas are obtained, as with ellipses, by fixing two points (foci ) and by
imposing the condition
d1 d2 = l |d1 d2 | = l
where l is a constant. (See the figures.)
Case 1. The foci are on the X -axis.
|d1 d2 | = l
( l = 2a , see t

Lecture 32 Theory Questions
1) As + and as 0. Use this to answer the following:
a) as +
i) ?
ii) (1) ?
iii) + 1 ?
b) as
i) ?
ii) (1) ?
iii) + 1 ?
2) The graph of () has a horizontal asymptote = 5.
a) What is the horizontal asymptote of the graph of () +

Lecture 37 Theory Questions
1. Which of the following systems of equations are linear?
a. 2 + 2 = 2
+ =2
b.
1
+ = 2
+ =3
c. + 2 = 4
=2
d. 23+4 + = 0
+ =1
2. True or False. Explain your answer.
a.
A system of two linear equations can have no solution.
b.

Lecture 35 Theory Questions
1. Use the definition of exponential equations to determine whether each of the following
is an exponential equation. In each case, explain your reasoning.
a. 2 = 9
b. 2 = 7
c. 2 = 3
d. 2 + 6
e. = 3
f.
25 =
g. () = 3
2. Find t

Lecture 42 Theory Questions
1. Let A be a 2x3 matrix and B be a 3x4 matrix. Explain why .
2. Does each of the following matrices have an inverse? If so, find its inverse and verify
that
1 = = 1 .
2
a. [
1
2
b. [
3
5
]
3
4
]
6
3. Does the identify matrix

Lecture 45d Theory Questions
1. Describe how the graphs of the hyperbolas 2 2 = 1 and 2 2 = 1 would differ.
2
2. Suppose the hyperbola 2
2
1
= 1 has a large value for a. Describe how the graph
would appear.
3. Write an equation of a hyperbola with the fo

Lecture 39 Theory Questions
1. For each of the reduced row echelon form of the augmented matrix of a system below,
a. State the number of solutions of the system
b. State the solution(s), if any.
1
i. [0
0
1
ii. [0
0
1
iii. [0
0
1
iv. [0
0
0
1
0
0
0
0
0
0

Lecture 45
Cramers Rule
(
Ex : Consider the system
to eliminate y :
(
ax + by = r d
ax + by = r
cx + dy = s
(
=
and apply elimination procedure
adx + bdy = rd
cx + dy = s (b)
cbx bdy = bs
a
r b
c
s d
rd bs
. Likewise, y =
=
= x =
a b
a
ad bc
c d
c
C

Lecture 45e Theory Questions
1. Based upon the conditions provided for each part, classify the curves that satisfy the
equation, 2 + 2 + = 0:
a. = > 0, < 0
b. > 0, < 0, < 0
c. < 0, < 0, < 0
d. = 0
e. = 0
(Please continue with the Lecture 45e Problems on t

Lecture 36a Theory Questions
1. Suppose that a quantity triples for every time interval t. Is this an example of
exponential growth? (Hint: replace 2 with 3 and repeat the derivation found in
lecture #36a).
2. Does the equation, () = (0) , describe expone

Lecture 43
(
Matrix Equations
x + 2y = 5
1 2
This can be written in matrix form. Let A =
3x y = 1
3 1
x
5
be the coefficient matrix of the system and X =
, B=
. Then the
y
1
system is equivalent to the single matrix equation
AX = B
1
1
Suppose we know

Lecture 41 Theory Questions
1. Let the dimension of A be 3x1, the dimension of B be 3x2, the dimension of C be 3x3 and
the dimension of D be 2x3.
Which of the following products are defined? Explain.
a.
b.
c.
d.
e.
f.
g.
2. Let = [
0 0
2 3
], B= [
]

Lecture 36b Theory Questions
1. When a quantity () decreases by a factor of
1
5
for every time interval T, does ()
experience exponential decay?
1
2. Recall that 2 = 21 . Use this fact to explain the relationship between exponential growth
and decay in ()

Lecture 44
Determinants
Recall :
1
a b
d b
1
If A =
and ad bc 6= 0, then A =
c d
ad bc c a
The number ad bc is a particular example of the determinant.
a b
Def : The determinant of a 2 2 matrix A =
is the number adbc .
c d
a b
, |A|, det A are possible n

Lecture 42
Properties of Operations with
Matrices. The Inverse Matrix
Because addition, subtraction and multiplication of a matrix by a number
reduce to the same operations performed with each entry, the usual algebraic
properties (commutative, associativ

Lecture 45a Theory Questions
1. By definition, which of the following are quadratic equations in two variables?
a.
2 + 2 + 2 = 0
b. 3 + 3 2 = 0
c. 2 2 + 3 =
d. 3 6 + 3 =
e. 2 + 2 = 1
2. Does the equation = 2 define a quadric curve? If so, rewrite = 2 in

Lecture 36 Theory Questions
1. Use the definition to determine whether each of the following is a logarithmic equation.
In each case, explain your reasoning.
a. 2 () = 2
b. ( 2 + 6 4) = 5
c. 5 = ( 2 + 6 4)
d. 3 = 2
e. 3 = ln(2)
f.
2 + 1 = log(2)
g. () = l

Lecture 34 Theory Questions
1. Let = 4, = 4, and = 2. Show that ( + ) () + ()
2. Let = 9, = 3, and = 3. Verify that () = () () for these values.
3. Without a calculator, evaluate 27 (81) using a change of base formula. (Hint: find an
appropriate base).
4.

Lecture 31 Theory Questions
1) Why does the definition of an exponential function require that the base be not
equal to 1?
2) Are the following exponential functions?
a) = 2
b) = 0.01
c) = (2)
d) = 5
e) = (2)
3) For which value of is = 1 independent of th

Lecture 43 Theory Questions
1. Consider the system of equations below:
2 + 2 = 4
3 4 = 1
Find A, B, and X to rewrite the system as a single matrix equation AX = B
A=
B=
X=
2. Let = [
1 2
3
], = [ ], X= []
3 3
5
Write the matrix equation AX = B as a system

Lecture 44 Theory Questions
1. Suppose that the first row in a square matrix consists of zeros. What is the determinant
of the matrix?
2. What is the determinant of an identity matrix?
3. Consider the following diagonal matrix:
1
[0
0
0
0
2
0
0
0
0
3
0
0

Lecture 45f Theory Questions
1. Rate the following statements as True or False. Provide an example or a counterexample
(Hint: You can use graphs).
a. A system of non-linear equations cannot have exactly two solutions.
b. A system of non-linear equations c

Lecture 45b Theory Questions
1. Describe how an ellipse would look like if its eccentricity is 0.
2. Describe how an ellipse would look like if its eccentricity is close to 1.
3. If the foci of an ellipse are on the y-axis, which semi-axis of the ellipse

Lecture 45 Theory Questions
1. Explain why Cramers rule is only applicable to a system where the number of equations
is equal to the number of variables.
2. Is Cramers rule applicable if the determinant of the system is equal to 0? Explain.
3. Consider th

Lecture 45b
Ellipse
Quadric curves can be defined geometrically in terms of distances between
points.
In particular, using two nails , a rope
and a pencil we can draw an ellipse:
Note : d1 + d2 is the same for all points on the ellipse. It is simply the
l