Solve 4 questions of quiz. each of them gives 0.25 point

1. Show that the following sets of elements in R³ form subspaces. (a). The set of all (x, y, z) such that x − 2y + z = 0. (b). The set of all (x, y, z) such that x = 3z and y = z.

2. (a). Let U = {(x, y) ∈ R² : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1} and W = {(x, y) ∈ R² : x 2 + y 2 ≤ 1}. Are these sets subspaces of R^2? (b). Find the sum U + W.

3. If U and W are subspaces of a vector space V, show that U + W is a subspace

4.Show that functions f(t) = t and g(t) = 1/t defined for t > 0 are linearly independent.

The height of a rock thrown upwards on the planet Mars with an initial velocity of 60 m/s is 2 approximated by h(?) = 60? − 1.86? . Fill in the average velocity over each time intervals shown in the table. Show all work for each calculation. Then approximate the instantaneous velocity of the rock after 10 seconds using the table of values.(?h?? ??? ???? ???? ???h ???????????; ?? ??? ????? ??? ???????. )

find the equation of tangent line to the curve r=1+3sinθ when θ=π/3

Let f(x) = cosh(x) and g(x) = sinh(x), a = 0 and b = 1.

A) Find the volume of the solid with base on the xy plane, bounded by the region above, whose cross-sections perpendicular to the x axis are squares.

B) Find the volume of the solid formed if the region above is rotated about the line y = 4.

C) Find the volume of the solid formed if the region above is rotated about the line x = 2.

Sketch the graph of the curve ? = 1 + sin ? for 0 ≤ ? ≤ 2? ? b) Find the slope of the tangent line to this curve at ? = 4 . c) Find the polar coordinates of the points on this curve where the tangent line is horizontal.

1. Find answers that meet the first-order optimization conditions, identify the shape of the function graph to determine whether these answers are maximum value or minimal value.

(a) f(x) = e^x + e^-x

(b) f(x) = x^3/(x+1)

(c) f(x) = x/lnx

2. Find answers that meet the second-order optimization conditions, and determine whether these answers are maximum value or minimal value.

(a) f(x) = e^x + e^-x

(b) f(x) = x^3/(x+1)

(c) f(x) = x/lnx

3. Determine whether each function is concave or convex. If the entire function is not concave or convex, look for concave and convex sections.

For the following u(x, y), show that it is harmonic and then find a corresponding v(x, y) such that f(z)=u+iv is analytic.

u(x, y)=(x^2-y^2) cos(y)e^x-2xysin(y)ex

The position of a particle moving along a line (measured in meters) is s(t) where t is measured in seconds. Answer all parts, include units in your answers.

s(t)=2t^3 +6t^2 −48t+7 −10<t<10

(a) Find the velocity function.

(b) Find all times at which the particle is at rest.

(c) On what interval is the particle moving to the right?

(d) Is the particle slowing down or speeding up at t = −1 seconds?

In the new Mission Impossible movie, Ethan Hunt must save the world from an evil genius who is going to release a computer bug that will shut down the whole power grid. In an attempt to creative and critically think through how to stop this evil genius, Ethan Hunt finds himself on the London Eye, Great Britain's largest Ferris Wheel. The Eye is centered at (0,75) with a radius of 60 meters. It makes one revolution every 24 minutes. The London Eye loads from bottom.

a) Write a set of parametric equations for the London Eye.

b) If it Ethan Hunt 60 minutes on the London Eye to figure out a plan, how far did he travel?

c) After Ethan Hunt figures out his plan, Agent Luther Stickell joins him on the Ferris Wheel. When is Agent Luther Stickell at a height of 100 meters?

Find the surface area of revolution about the x-axis of y = 5 sin ( 5 x ) over the interval 0 ≤ x ≤ π/5

A savings bond offers interest at a rate of 8.8% compounded semi-anually. Suppose that a $1500 bond is purchased.

a) Determine the value of the investment after 12 years.

b) Describe how the shape of the graph of this function would change if a bonus of 5% of the principal was added after 5 years had passed

c)Describe how the shape of the graph changes if the size of the initial investment was doubled.

What does it means for a set of vectors a1; a2; : : : ; ; an to be linearly independent?
What is the span of the set?

This question is about linear algebra

1. compute the least non-negative residue of 4^n (mod 9) for n=1,2,3,4,5.... prove that 6*(4^n)=6 (mod 9) for every n>0.

2. find nice tests for divisibility of numbers in base 34 by each of 2,3,5,7,11,and 17.

3. in Z/15Z, find all solutions of : (i) [36]X=[78]. (ii) [42]X=[57] (iii) [25]X=[36]

4. in Z/26Z, find the inverse of [9], [11], [17], and [22]

4. write the set of solutions of x=5 mod24. x=17 (mod 18)

for all equation line, there are triple line.

MINIMIZATION BY THE SIMPLEX METHOD

convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

1>.

Minimize z = 6x₁ + 8x₂

subject to 2x₁ + 3x₂ ≥ 7

4x₁ + 5x₂ ≥ 9

x₁, x₂ ≥ 0

2>.

Minimize z = 4x₁ + 3x₂

subject to x₁ + x₂ ≥ 10

3x₁ + 2x₂ ≥ 24

x₁, x₂ ≥ 0