#ATTENDANCE QUIZ for Lecture 11 of Math251(Dr. Z.)
#EMAIL  RIGHT AFTER YOU WATCHED THE VIDEO
#BUT NO LATER THAN  Oct. 12, 2020, 8:00PM (Rutgers time)
#EXTENDED TO Oct. 17
#THIS .txt FILE (EDITED WITH YOUR ANSWERS)
#TO:
#DrZcalc3@gmail.com
#Subject: aq11
#with an ATTACHMENT CALLED:
#aq11FirstLast.txt
#(e.g. aq11DoronZeilberger.txt)




#LIST ALL THE ATTENDANCE QUESTIONS FOLLOWED BY THEIR ANSWERS

1. Question: Who proved that every positive integer can always be written as a sum of four squares 9=3^2 10=3^2 + 1^2?

Answer: Joseph LaGrange

2. Question: Use Lagrange multipliers (no credit for other methods) to find the largest value that x +3y + 5z can be,
given that x^2 + y^2 + z^2 = 35.

Answer:

f_x = 1
f_y = 3
f_z = 5

grad(f) = <1,3,5>

g_x = 2x
g_y = 2y
g_z = 2z

grad(g) = <2x, 2y, 2z >

grad(f) = L * grad(g)
<1,3,5> = L * <2x, 2y, 2z >

1 = L*2x  => x = 1/2L
3 = L*2y  => y = 3/2L
5 = L*2z  => z = 5/2L

15 = 8xyzL^3

Solving for x: x = 15 / 8yzL^3   x = 1 / 4 L

Solving for y: y = 15 / 8xzL^3   y = 3 / 8 L

Solving for z: z = 15 / 8yxL^3   z = 5 / 8 L

Solving for L: 1/16L^2 + 9/64L^2 + 25/64L^2 = 35   => L = 0.1302 and L = -0.1302


f(0.0325, 0.0488, 0.0814) = 0.5859 

The largest value can be 0.5859. 

3. Question:**Use Maple: Let [i]= the ith digit of your RUID, if it is zero make it 5. Find the maximum value 
of a[1]x^2 + a[5]xy +a[9]y^2 ont he curve a[5]x^3 +a[7]xy +a[5]y^3 = a[5]

Answer: Not sure how to complete, I am trying to practice more with Maple and I feel that when I input the function, there isn't an output. 

4. Question: Finish this up!

Answer: 

y = 1.639
x = 2.449

f(1.639, 2.449) = 64.72