7.7: Determinants
y
l2=a2+c2
1
For 2 2: (assume a = 0)
A=
a b
c d
a
b
0 d bc/a
a(d bc/a) = ad bc = 0
Def.: The determinant of A is
def
det(A) = ad bc
Geometric Interpretation:
|det(A)| is the area of the parallelogram generated by the two
column (or row)
4.7: Forced Harmonic Motion
Periodically forced mass-spring system: mx +x +kx = F0 cos t
2
or x + dx + 0 x = A cos t
where d = /m, 0 = k/m, A = F0/m
(A) Undamped Case
2
x + 0 x = A cos t
(1) Beats. Assume IC: x(0) = 0, x (0) = 0
2
Try particular solution:
7.5 Span of a Set of Vectors
Def.: Given vectors x1, . . . , xk Nullspaces:
in Rn , the set of all their linear Ex.: Consider Ax = 0 for
1
3 2
combinations is called their
0
A = 0 1
span, and is denoted by
0
0
0
span(x1, . . . , xk )
Free variable: x3 =
7.6 Square Matrices
A: square matrix (n n)
Def.: A is nonsingular if for
any b, Ax = b has a solution
Thm.: The following properties are equivalent:
A is nonsingular
REF (A) has no free variables
RREF (A) = I
(identity matrix)
Ax = b has a unique
solu
8.2-3: Geometric Interpretation and Qualitative Analysis
Autonomous system:
x = f (x),
x = [x1, . . . , xn]T
Example: Lotka-Volterras
predator-prey equations
R = (a bF )R
F = (c + dR)F
For any t: x(t) Rn
Rn : phase space
(n = 2: phase plane)
a, b, c, d >
Name:
Second midterm practice exam, Math 340
Each problem is worth 10 points. One problem with the lowest score will be
dropped. Good luck!
Problem
1
2
3
4
5
6
7
8
Total/Total - lowest
Score
1. (10 pts) Find the basis for the nullspace of the following ma
Name:
Sections 1.5 & 1.6 : Math 115
Answer the following
Section 1.5
1. Solve for x if
|2x 3| = 9.
2. Solve for w if
w2 = 3(w + 6)
3. Solve for x if
1 1
1
= +
y
x z
Section 1.6 - How to succeed with word problems
1. Find out whats being asked. What is the
Untitled
MA115 : Section 2.2
Warm-up Problem Find the equation of the line
Functions
1. To graph piecewise functions, simply think of the function as two different functions and graph each one over ITS GIVEN DOMAIN.
MA115 : Section 2.2
4. The graph of a p
Untitled
MA115 : Section 2.1 & Variation
Warm-up Problem A spoonful of cream is taken from a pitcher of
cream and put into a cup of coee. The coee is stirred. Then a spoonful
of this mixture is put into the pitcher of cream. Is there now more cream in
the
MA115 : Section 2.2 : Average Rate of Change
Warm-up Problem Could this curve denotes temperature of a cup of
coee at a given time, where time is plotted along the x-axis?
1. The function f is increasing on an interval (a, b) if
x1 < x2 f (x1 ) < f (x2 )
Name:
Sections 1.1 and 1.2 Exercises : Math 115
Answer the following
1. Find the indicated sets if
A = cfw_1, 2, 3, 4, 5, 6, 7, B = cfw_2, 4, 6, 8 and C = cfw_7, 8, 9, 10
(a) A B
(b) A B
(c) B C
(d) A B C
2. Find the indicated sets if
A = cfw_x|x 2, B = c
Untitled
Name:
Sections 1.5 & 1.6 : Math 115
Answer the following
Section 1.5
1. Solve for x if
|2x
3| = 9.
2. Solve for w if
w2 = 3(w + 6)
3. Solve for x if
1
1 1
= +
y
x z
Section 1.6 - How to succeed with word problems
1. Find out whats being asked. Wh
MA115 - 3.1-3.3
MA115 : Sections 3.1 & 3.2 & 3.3 Polynomials, Long Division &
Rational Zeros
1. Find a polynomial of degree 3 that has zeros 2, 1, 3
2.
1
is a rational function with NO zeros.
x
3. We can divide polynomials
x + 5|x2
3
- 30 -
2x
35
MA115 -
MA115 2.7
MA115 : Section 2.7 : Combining Functions
The most important way to combine functions is by composition of
functions.
You basically apply each function rule, one after the other.
For example, if f (x) = x2 and g(x) = 2x + 1, we write
h(x) = f (g
MA115 - 2.4
MA115 : Section 2.4 : Function Transformations and Max/Min
Sketch the transformations indicated in each caption. Scale the axes as
appropriate.
(a) 4 Vertical Shifts of f (x) = x6
(b) 4 Horizontal Shifts of f (x) = x3
(c) 4 Vertical Stretch/Sh
Untitled
MA115 : Section 2.2
Warm-up Problem Find the equation of the line
Functions
1. To graph piecewise functions, simply think of the function as two different functions and graph each one over ITS GIVEN DOMAIN.
2. Sketch the graph of the function
2x
MA104 Final Exam Practice
1. Solve the following triangle: A = 65 , B = 15 , C = 100
2. Solve the following triangle: A = 25 , b = 10, c = 0
3. Solve the following triangle: a = 8, b = 12, A = 40
4. Solve the equation for x
9(x + 4) +
5 = 5(x 1)
45
5. Yo
1. Simplify each rational expression (Section 4.1)
a.
b.
c.
(
(
)
)(
(
)
)
d.
(
(
)(
)(
)
)
e.
(
(
2. Express in simplest form (Section 4.2)
a.
)
)(
)
b.
(
)(
) (
(
)
)
c.
(
(
)(
)
)(
(
(
)
)
)
(
)
(
)
d.
(
)
(
)
3. Add or subtract as indicated. Express y
MA 104 1/14/14
Chapter 1 - Review
Name_
To receive credit, you must show your work.
1. Evaluate WITHOUT a calculator.
a.
b.
(
(
)
)
( )
c.
d.
(
)
(
)
(
)(
e.
f.
[
(
[
]
)
]
( )
[
]
[
]
MA 104 1/14/14
Chapter 1 - Review
Name_
2. Simplify by combining like
MA 104 1/21/14
Chapter 2 Word Problems
Name_
1. Find a number such that one-half of the number is the same as 10 more thanone-fifth of the number.
[
[
]
[
[
]
]
]
[
]
2. Suppose that the width of a rectangle is 3 centimeters less than two-thirds of its le