Let A be an n × n matrix which is not 0 but A² = 0. Let I be the identity matrix.

a)Show that A is not diagonalizable.

b)Show that A is not invertible.

c)Show that I-A is invertible and find its inverse.

Use ‘Reduction of Order’ to find a second solution y2 to the given ODEs:

(a) y′′+2y′+y=0, y1 =xe−x

(b) y′′+9y=0, y1 =sin3x
(c) x2y′′+2xy′−6y=0, y1 =x2

(d) xy′′ +y′ =0, y1 =lnx

Solve a system of equations:
1-
2x = 5 mod 15  
3x = 1 mod 4

2-
x = 5 mod 15
x = 2 mod 12

(Hint: Note that 15 and 12 are not relatively prime. Use the Chinese remainder
theorem to split the last equation into equations modulo 4 and modulo 3)

Let W be the set of P₄ consisting if all polynomials satisfying the conditions p(-2)=0.

a.) prove that W is a subspace of P₄ by checking all 3 conditions in the definition of subspace.

b.) Find a basis for W. Prove that your basis is actually a basis for W by showing it is both linearly independent and spans W

c.) what is the dim(W)

For the wheel W₉ find the minimal coloring. Show that the coloring is sufficently large by enumnerating it. Explain why it may not be colored with fewer.

Problem 7.3. Let f (x, y) = x6 + 3xy + y2 + y4.
(a) Show that f remains unchanged if you replace x by −x and y by −y. Hence,
if (x, y) is a critical point of f, so is (−x, −y). Thus, critical points other than
(0, 0) come in ± pairs.
140 7 Optimization in Several Variables
(b) Compute the partial derivatives of f . Show that solve applied directly to
the system fx = fy = 0 fails to locate any of the critical points except for (0, 0).
(c) Let’s compensate by eliminating one of the variables and then using solve
followed by double. First solve for y in terms of x in the equation fx = 0.
Substitute back into the formula for fy and then apply first solve and then
double. You should end up with three critical values of x, giving a total of
three critical points. Find the numerical values of their coordinates. (Be sure
you have set x and y to be real; otherwise you will also end up with many
irrelevant complex critical points.)
(d) Confirm the calculation of the critical points by graphing the equations fx =
0 and fy = 0 on the same set of axes (using fimplicit and hold on). You
should see exactly one additional pair of critical points (in the sense of (a)).
(e) Classify the three critical points using the second derivative test.
(f) Apply fminsearch to f with the starting values (1, 1) and (0, 0). Show
that in the first case you go to a minimum and that in the second case you stay
near the saddle point.

Problem 7.1. Let f (x, y) = x4 − 3xy + 2y2.
(a) Compute the partial derivatives of f as well as its discriminant. Then use
solve to find the critical points and to classify each one as a local maximum,
local minimum, or saddle point.
(b) Check your answer to (a) by showing that fminsearch correctly locates
the same local minima when you start at (0.5, 0.5) or at (−0.5, 0.5).
(c) What happens when you apply fminsearch with a starting value of
(0, 0)? Explain your answer.
(d) What are the values of f at the extrema? Now, using fmesh, graph the function
on a rectangle that includes all the critical points. Experiment with view
and axis until you get a picture that shows the behavior near the critical points.
Use the graph and all the previous data to justify the assertion: Sometimes symbolic
and/or numerical computations are more revealing than graphical information.

What is an equilibrium solution? I am in an Elementary Differential Equations course. The problem is dy/dx = e^y-1; (0,0) and (1,1) and wants the equilibrium solutions. Been a few years since I've taken a math course I'm not finding any good explainations for what it is, thanks.

Ken played golf yesterday and shot 107. Considering that he normally shoots in the low 80’s or high 70’s, this round of golf really frustrated him. It was so frustrating that he decided to buy new golf clubs. But first he had to give his old golf clubs away. He gave half of his golf clubs, plus half a club more, to Daniel. Then he gave half of his remaining golf clubs to Gary. Then he gave half of his remaining golf clubs and half a club more to Will. This left Ken with one club (his putter), which he decided to keep. How many golf clubs did Ken start with before giving them away?

Must be answered by working backwards. No algebra.

Problem 7-19 (Algo) A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the urn follow a Poisson distribution at the rate of 2.5 per minute. In serving themselves, customers take about 16 seconds, exponentially distributed.

a. How many customers would you expect to see, on average, at the coffee urn? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. How long would you expect it to take to get a cup of coffee? (Round your answer to 2 decimal places.)

c. What percentage of time is the urn being used? (Do not round intermediate calculations. Round your answer to 1 decimal place.)

d. What is the probability that three or more people are in the cafeteria? (Do not round intermediate calculations. Round your answer to 1 decimal place.)

e. If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of 16 seconds, how many customers would you expect to see at the coffee urn (waiting and/or pouring coffee)? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

f. If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of 16 seconds, how long would you expect it to take (in minutes) to get a cup of coffee, including waiting time? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

A ball is thrown vertically upward from the top of a building 112 feet tall with an initial velocity of 96 feet per second. The distance s  (in feet) of the ball from the ground after t seconds is s (t) = 112 + 96t - 16t2 Complete the table and discuss the interpretation of each point. t s(t) Interpretation 0 0.5 1 2 100 100 200 200 Answer these questions. After how many seconds does the ball strike the ground? After how many seconds will the ball pass the top of the building on its way down? How long will it take the ball to reach the maximum height? What is the maximum height?

The question is to use Matlab to find the clamped cubic spline v(x) that interpolates a function f(x) that satisfies: f(0)=0, f(1)=0.5, f(2)=2, f(3)=1.5, f'(0)=0.2, f'(3)=-1 and then plot v(x).

This is my code so far:

x = [0 1 2 3];

y = [0 0.5 2 1.5];

cs = spline(x,[0 y 0]);

xx = linspace(0,3,101);

figure()

plot(x,y,'o',xx,ppval(cs,xx),'-');

IS THIS RIGHT? HOW CAN I GET MATLAB TO GIVE ME THE EQUATION OF v(x)?

1) Give parametrizations of the following circles with the indicated centers C(a, b) and radius R, and any other indicated properties.

a) C(1, 1), R = 2, traversed once counter clockwise.
b) C(1, 1), R = 2, traversed once clockwise.
c) C(0, 0), R = 1, starts at (0, 1).
d) C(0, 0), R = 1, starts at (−1, 0), traversed once clockwise.
e) C(0, 0), R = 1, traverses the circle twice on the domain 0 ≤ t ≤ 1.