Let x be a set and let R be a relation on x such x is simultaneously reflexive, symmetric, and antisymmetric. Prove equivalence relation.

. For this problem, you may find Proposition 3.1.5 useful which in turn implies that tan x is continuous whenever x is not an odd multiple of π 2 . Moreover, you can assume that sin x and cos x are positive and negative on appropriate intervals. Let I := (− π 2 , π 2 ). (a) Show that tan x is strictly increasing and, hence, injective on I. (b) Prove that lim x→− π 2 + tan x = −∞ and lim x→π 2 − tan x = ∞ Use this to conclude that f(I) = R. You may find Exercise 6 in Section 3.7 useful here. (c) Show that arctan x = tan−1 x maps R to I and is differentiable everywhere with d dx tan−1 x = 1 1 + x 2 for all x ∈ R. (d) Prove that limx→∞ tan−1 x = π 2 and lim x→−∞ tan−1 x = − π 2

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

(62). (Transient Behavior of a Mixing tank): A tank initially holds 80 gal of a brine solution containing 0.125 lb of salt per gallon. At t=0, another brine solution containing 1 lb of salt per gallon is poured into the tank at a rate of 4 gal/min, while the well-stirred mixture leaves the tank at a rate of 8 gal/min. (a) Formulate a model for describing the transient behavior of the tank (b) Find the amount of salt in the tank when the tank contains exactly 40 gal of the solution.

Solve by solving the dual problem.

Minimize

z = 30x₁ + 15x₂ + 28x₃,

subject to

A mass weighing 8 lb is attached to a spring hanging from the ceiling, and comes to rest at its equilibrium position. The spring constant is 4 lb/ft and there is no damping.

A. How far (in feet) does the mass stretch the spring from its natural length?
L=  

B. What is the resonance frequency for the system?
ω0=  

C. At time t=0 seconds, an external force F(t)=3cos(ω0t) is applied to the system (where ω0 is the resonance frequency from part B). Find the equation of motion of the mass.
u(t)=

D. The spring will break if it is extended by 5L feet beyond its natural length (where L is the answer in part A). How many times does the mass pass through the equilibrium position traveling downward before the spring breaks? (Count t=0 as the first such time. Remember that the spring is already extended L ft when the mass is at equilibrium. Make the simplifying assumption that the local maxima of u(t) occur at the maxima of its trigonometric part.)
times.

CSM Machine Shop is considering a four-year project to improve its production efficiency. Buying a new machine press for $415,000 is estimated to result in $154,000 in annual pretax cost savings. The press falls in the MACRS five-year class (MACRS Table) and it will have a salvage value at the end of the project of $55,000. The press also requires an initial investment in spare parts inventory of $16,000, along with an additional $3,000 in inventory for each succeeding year of the project. The shop’s tax rate is 25 percent and its discount rate is 12 percent. Calculate the project's NPV. (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Discrete math problem:

Prove that there are infinitely many primes of form 4n+3.

find the coefficient for the terms

(A) What is the coefficient for the term x 4y 3 in (x + y) 7 ?

(B) What is the coefficient for the term x 4y 3 in (x − y) 7 ?

(C) What is the coefficient for the term x 2y 3 z 2 in (x + y + z) 7 ?

1. Prove that the Cantor set contains no intervals.

2. Prove: If x is an element of the Cantor set, then there is a sequence X_n of elements from the Cantor set converging to x.

Number Theory

Exercise 1 Prove that the equation 3x^2 + 2 = y^2 has no solution (x, y) ∈ Z × Z. (Hint: consider the associated congruence modulo 3.)

Exercise 2 Prove that the equation 7x^3 + 2 = y^3 has no solution (x, y) ∈ Z × Z. (Hint: consider the associated congruence modulo 7.)

Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) = (2,3,-5) and T( e⃗2 ) = (-1,0,1).

1. Determine the standard matrix of T.

2. Calculate T( ⃗u ), the image of ⃗u=(4,2) under T.

3. Suppose T(v⃗)=(3,2,2) for a certain v⃗ in R2 .Calculate the image of ⃗w=2⃗u−v⃗ .

4. Find a vector v⃗ inR2 that is mapped to ⃗0 in R3.

7. Show that the dual space H' of a Hilbert space H is a Hilbert space with inner product (', ')1 defined by
(f .. fV)1 = (z, v)= (v, z), where f.(x) = (x, z), etc.

I would like to know how to plot two graphs in matlab and find where they intercept.