#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

#Question 1:
# Who proved that every possible integer can always be written as a sum of squares

Answer: Joseph Louis Lagrange



Question 2:
# Using Largange multipliers (no credit for other methods) to find the largest value that x+3*y+5*z can be given that x^2+y^2+z^2=35

Answer: 
f(x,y,z)= x+3*y+5*z
g(x,y,z)= x^2+y^2+z^2-35

Find gradients:
f_grad= <f_x,f_y,f_z>= <1,3,5>

g_grad= <g_x,g_y,g_z>= <2x,2y,2z>

Apply f_grad= λ* g_grad
Yields equations:
1=2xλ
3=2yλ
5=2zλ
Solve these equations x,y, and z in terms of λ and plug equations into g(x,y,z) then simply to get:
λ= -1/2,+1/2

To get maximum plug =+1/2 in to get x,y, and z components:

Maximum is (1,3,5)

Question 3: Let a[I]:= the I-th digit of your RUID, if it is zero make it 5
Find the maximum value fo a[1]*x^2+a[4]*x*y+a[9]*y^2 on the curve a[5]*x^3+a[7]*x*y+a[5]*y^3=a[5]

Answer: 
f(x,y)= x^2+5*x*y+2*y^2
g(x,y)= 5x^3+7*x*y + 5*y^3=5


f_grad= <f_x,f_y>= <2x+5y,5x+4y>
g_grad= <g_x,g_y>= <15*x^2+7*y,7*x+15*y^2>

Apply f_grad= λ* g_grad
Yields equations:
2x+5y=15x^2*λ +7y*λ
5x+4y=7x*λ + 15y^2* λ

Simplify to obtain:
5x^3-14x^2=75y^3+2y^2-60x^2*y-5*x*y