Two people are on a city block. Person A is on the northeast corner and Person B is on the southwest corner. Person A starts walking towards the southeast corner at a rate of 3 ft/sec. Four seconds later Person B starts walking towards the southeast corner at a rate of 2 ft/sec. At what rate is the distance between them changing (a) 10 seconds after Person A starts walking and (b) after Person A has covered half the distance?

Suppose u(t,x)solves the initial value problem Utt = 4Uxx + sin(wt) cos(x),

u(0,x)= 0 , Ut(0,x) = 0. Is h(t) = u(t,0) a periodic function?

(PDE)

True or False and support your decision

1.) if Ax=0 has only one trivial solution, then Ax=0 has a unique solution.

2.)The linear system given by Ax=0 consisting of 3 equations and 4 unknowns has infinitely many solutions

3.) If the vector set {v,u,w} is linearly independent, then w is a linear combination of u and v

4.) If A is a 3by5 matrix with 3 pivots,then the columns of A span R5

Find the smallest n ∈ N such that 2(n + 5)^2 < n^3 and call it n^0，Show that 2(n + 5)^2 < n^3 for all n ≥ n^0.

Find at least the first four nonzero terms in a powerseries expansion about x = 0 for a general solution to thegiven differential equation: Include a general formula for the coefficients (recurrence formula). x=0; (x^2 +4)y'' + y=x

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum of the positive divisors of n (as in the notes).

(a) Evaluate τ (1500) and σ(8!).

(b) Verify that τ (n) = τ (n + 1) = τ (n + 2) = τ (n + 3) holds for n = 3655 and 4503.

(c) When n = 14, n = 206 and n = 957, show that σ(n) = σ(n + 1).

The Janie Gioffre Drapery Company makes three types of draperies at two different locations. At location I, it can make 10 pairs of deluxe drapes, 20 pairs of better drapes, and 13 pairs of standard drapes per day. At location II, it can make 20 pairs of deluxe, 50 pairs of better, and 6 pairs of standard per day. The company has orders for 3000 pairs of deluxe drapes, 6300 pairs of better drapes, and 1800 pairs of standard drapes. If the daily costs are $450 per day at location I and $600 per day at location II, how many days should Janie schedule at each location to fill the orders at minimum cost?

location I days ?

location II days?

Find the minimum cost. $

Show that the integral ( over a volume) of the curl of the vector A is equal to the integral over a closed surface (containing the volume) of A x da

Let x ∈ Rⁿ be any nonzero vector. Let W ⊂ R^nxn
consist of all matrices A such that Ax = 0. Show that W is a subspace and find its dimension.

Check that the following differential equation is not exact. Find an integrating factor that makes it exact and solve it.

ydx + (3 + 3x-y) dy = 0

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Solve the initial value problem:

(cos xsen x-xy ^ 2) dx + (1-x ^ 2) ydy = 0 if y (0) = 2

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Solve the initial value problem:

y ^ (2) cosx dx + (4 + 5ysenx) dy = 0; y (π / 2) = 1

Find the particular integral of the following differential equations.(Explain each step clearly)

(a) d²y/dx² + y = (x + 1) sin x. show that the answer is yp(x) = − 1/8 [ (2x² + 4x − 1) cos x − (2x + 2) sin x ]

(Hint:In this case, we substitute sin αx or cos αx with e^iαx then use the shift operator. In the case of sin αx we extract the imaginary part.)

How many permutations of the letters ABCDEFGHIJKLM do not contain the strings “BAD” or “DIG” or “CLAM” consecutively? (Hint: Inclusion-exclusion and subtraction).

**I KNOW THE ANSWER IS NOT 13! - 11! - 11! - 10! ** - please do not give that as one.

2. (a.) Consider [3]₁₂ in Z₁₂. Is [3]₁₂ a zero divisor of Z₁₂ and does [3] have a multiplicative inverse in Z₁₂? Justify your answer.

(b.) Prove that [3]₁₀ has a multiplicative inverse in Z₁₀ and determine the order of [3] in Z^x₁₀. Justify your answer.

(c.) Prove that [6]₈ is a nilpotent element of Z₈. Justify your answer.