#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

#FIRST ATTENDANCE PROBLEM  OF LECTURE 11
#WHO PROVED THAT EVERY POS. INTEGER CAN ALWAYS BE WRITTEN AS A SUM OF FOUR SQUARES?

#ANS.
#Joseph Lagrange

#ATTENDANCE QUESTION #2 FOR LECTURE 11
#Use Largange mutipliers(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

#ANS.
#fx=1	fy=3	fz=5
#grad(f)=<1,3,5>
#gx=2x	gy=2y	gz=2z
#grad(g)=<2x,2y,2z>
#λ<1,3,5>=<2x,2y,2z>
#{2x=λ,2y=3λ,2z=5λ,x^2+y^2+z^2=35}
#x=1,y=3,z=5,λ=2
#1+3*3+5*5=35
#THE ANSWER IS 35

#THIRD ATTENDANCE QUESTION
#Let a[i]:=The i-th digit of your RUID,if it is zero make it 5
#FIND THE MAXIMUMVALUE OF a[1]x^3+a[4]xy+a[9]y^2 ON the curve a[5]x^3+a[7]xy+a[5]y^3=a[5]

#ANS.
#f=2x^3+5xy+5y^2
#g=5x^3+2xy+5y^3=5
#fx=6x^2+5y	fy=5x+10y
#grad(f)=<6x^2+5y,fy=5x+10y>
#gx=15x^2+2y	gy=2x+15y^2
#grad(g)=<15x^2+2y,2x+15y^2>
#λ<6x^2+5y,5x+10y>=<15x^2+2y,2x+15y^2>
#{λ6x^2+λ5y=15x^2+2y,λ5x+λ10y=2x+15y^2,5x^3+2xy+5y^3=5}
#{ λ= 1.058976679, x = 0.5700821043, y = 0.8528193121}
#2x^3+5xy+5y^2=6.437935113
#THE ANSWER IS 6.437935113

#FOURTH ATTENDANCE PROBLEM
#FINISH THIS UP!

#ANS.
#y=sqrt(8/3)
#x=4/sqrt(8/3)
#f=48
#THE ANSWER IS 48