#ATTENDANCE QUIZ for Lecture 7 of Math251(Dr. Z.)
#EMAIL  RIGHT AFTER YOU WATCHED THE VIDEO
#BUT NO LATER THAN  Sept. 28, 2020, 8:00PM (Rutgers time)
#THIS .txt FILE (EDITED WITH YOUR ANSWERS)
#TO:
#DrZcalc3@gmail.com
#Subject: aq7
#with an ATTACHMENT CALLED:
#aq7FirstLast.txt
#(e.g. aq7DoronZeilberger.txt)




#LIST ALL THE ATTENDANCE QUESTIONS FOLLOWED BY THEIR ANSWERS

#Attendance Question #1:
#Let a:=5th digit of YOUR RUID; b:=2nd digit of YOUR RUID 
#Let f(x,y):= x^a + b*x^(2*a)*y^3
#Find THE TWO PARTIAL DERIVATIVES of f(c,y)
#f_x(x,y) , f_y(x,y)

#Answer: Since RUID digits 5 and 2 are both 0, I will use a:=6th:=6 and b:=1st:=2
#f_x(x,y) = 6x^5 + 24(x^11)(y^3)
#f_y(x,y) = x^6 + 6(x^12)(y^3)



#Attendance Question #2:
#With a and b as above, fix f_x (z(x,y)) and f_y (z(x,y)) IF x, y, z ARE RELATED
#BY THE RELATIONSHIP (EQUATION)
#x^a * y^2 * z^b + x^2 * y^(3*b) * z^3 = exp(x*y*z)

#Answer: Since RUID digits 5 and 2 are both 0, I will use a:=6th:=6 and b:=1st:=2
#f_x(z(x,y))=(y*(6*(x^5)*(z^2)+2*(x^6)*z)+(y^5)*(2*x*(z^3)+3*(x^2)*(z^2)))/(y*exp(x*y*z))
#f_y(z(x,y))=((x^6)*(2*y*(z^2)+2*(y^2)*z)+(x^2)*(6*y^5*z^3+3*(y^6)*(z^2)))/(x*exp(x*y*z))



#Attendance Question #3:
#Do the second part, find f_y(1,1)

#Answer:
#((x^2)+(y^2))' = (x*y*z + x*y*(z^5))'
#2*y = x*(y*z)' + x*(y*(z^5))'
#2*y = x*(z+(x*z')) + x*((z^5)+5*y*(z^4)*z')
#6z'=0
#z'=0



#Attendance Question #4:
#Is the answer the same, is f_y(1,1) also 0? Could you have predicted it without doing the #calculation?

#Answer:
#Yes both answers are 0. This could have been predicted because the placement and amounts #of x and y are equal in the equation so acting as if one is a constant will always result #in the same answer



#Attendance Question #5:
#Let a and b be as above
#Find the equation of the tangent plane to the surface z= x^a + y^b + a*b*x^y at
#the point (1,1,2+a*b)

#Answer: Since RUID digits 5 and 2 are both 0, I will use a:=6th:=6 and b:=1st:=2
#f_x=6*(x^5) + 12*y*(x^y)
#f_y=2*y
#f_x(1,1,14) = 18
#f_y(1,1,14) = 2
#z=18*x +2*y -6