Use this theorem to find the inverse of the given matrix or show that no inverse exists. (If an answer does not exist, enter DNE in any cell.)

1. Water with a small salt content (5kg per 1000 liters) is flowing into a very salty lake at the rate of 4 x 10⁵ li/hr. The salty water is flowing out at the rate of 10⁵ li/hr. If at some time (t=0) the volume of the lake is 10⁹ liters, and its salt content is 10⁷ kg, find the salt content at time, t. Assume that the salt is mixed uniformly with the water in the lake at all times.

Determine the number of DISTINCT colorings of the four faces of a tetrahedron, where each face is a color from the set { orange, purple, black} - Frobenius Orbit counting

4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?² = ?²}. Prove that ? is an equivalence relation, and concisely characterize how its equivalence classes are different from simple real-number equality.

1. Let {a_n} be a bounded sequence. In this question, you will prove that there exists a convergent subsequence.

Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, a_m > a_n for all n > m

1. (a) Suppose {a_n} has infinitely many crests. Prove that the crests form a convergent subsequence.
2. (b) Suppose {a_n} has only finitely many crests. Let a_n1 be a term with no subsequent crests. Construct a convergent subsequence with a_n1 as the first term.

Prove that if f is a bounded function on a bounded interval [a,b] and f is continuous except at finitely many points in [a,b], then f is integrable on [a,b]. Hint: Use interval additivity, and an induction argument on the number of discontinuities.

Make up a conditional statement of your own, clearly state it then find the following:

The statement:

What is its converse?

What is its inverse?

What is its contrapositive?

What is the sufficient condition of the statement?

What is the necessary condition of the statement?

how much horsepower would a car need to travel at the speed of sound at sea level?

assume the car weighs 4436 lb
assume no wind
assume no driveline horsepower loss

coefficients
Cd = .398
A = 26.72 ft^2
Crr = .015
p = .002377

Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rⁿ into an orthonormal basis. Use the vectors in the order in which they are given. On part D) Use the inner product <u, v> = 2u₁v₁ + u₂v₂ in R² and the Gram-Schmidt orthonormalization process to transform the vector.

A) B = {(24, 7), (1, 0)}

u1=____

u2=____

B)

B = {(3, −4, 0), (3, 1, 0), (0, 0, 2)}

u1=___

u2=___

u3=___

C)

B = {(−1, 0, 1, 2), (0, 1, 2, 2), (−1, 1, 0, 1)}

u1=___

u2=___

u3=___

D)

{(−2, 1), (2, 9)}

u1=___

u2=___

• Consider the group Z/mZ ⊕ Z/nZ, for any positive integers m and n that are divisible by 4. How many elements of order 4 does G have, and why?

4. Use the Euler Characteristic Theorem to prove that if G is drawn in the plane with n connected components, then IVI - |E| +|F| = n +1

Give examples of derangements of {1,2,3,4,5,6,7,8,9} of order 3 and of order 20. (A permutation in Sn is a derangement if it has no fixed points).

AllElectronics carries 1000 products, P1, . . . , P1000.

Consider customers Ada, Bob, and Cathy such that Ada and Bob purchase three products in common, P1, P2, and P3. For the other 997 products, Ada and Bob independently purchase seven of them randomly.

Cathy purchases 10 products, randomly selected from the 1000 products.

- In Euclidean distance, what is the probability that dist(Ada,Bob) > dist(Ada,Cathy)?

- What if Jaccard similarity is used?

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2
y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t

y′ = 5t4y, y(0) = 1, solution: y(t) = et5
y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3

For the IVPs above, make a log-log plot of the error of Backward Euler and Implicit Trapezoidal Method, at t = 1 as a function of hwithh=0.1×2−k for0≤k≤5.