#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 (0); b:=2nd digit of your RUID (9)
#LET f(x,y):= x^a + b*x^(2*a)*y^3
#Find THE TWO PARTIAL DERIVATIVES of f(x,y)

#ANSWER TO QUESTION 1:
#f(x,y) = x^0 + 9*x^(2*0)*y^3
#f(x,y) = 1 + 9*y^3
#f_x(x,y) = 9*y^3
#f_y(x,y) = 27*y^2



#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 TO QUESTION 2:
#z(x,y) = x^0*y^2*z^9 + x^2*y^(3*9)*z^3 = exp(x*y*z)
#z(x,y) = y^2*z^9 + x^2*y^27*z^3 = exp(x*y*z)
#f_x(z(x,y)) = y^2*[z^9]' + y^27*[x^2*z^3]' = [x*z]'*exp(x*y*z)
#f_x(z(x,y)) = y^2*9*z^8*z' + y^27*(2*x*z^3 + 3*z^2*z'*x^2) = (z+x*z')*exp(x*y*z)
#f_x(z(x,y)) = 9*y^2*z^8*z' + 2*x*y^27*z^3 + 3*x^2*y^27*z^2*z' 
#	     = x*z'*exp(x*y*z) + z*exp(x*y*z)
#f_x(z(x,y)) = 9*y^2*z^8*z' + 3*x^2*y^27*z^2*z' - x*z'*exp(x*y*z) 
#	     = z*exp(x*y*z) - 2*x*y^27*z^3 
#f_x(z(x,y)) = z'*(9*y^2*z^8 + 3*x^2*y^27*z^2 - x*exp(x*y*z)) = z*exp(x*y*z) - 2*x*y^27*z^3
#f_x(z(x,y)) = z'= (z*exp(x*y*z) - 2*x*y^27*z^3) / (9*y^2*z^8 + 3*x^2*y^27*z^2 - x*exp(x*y*z))
#f_y(z(x,y)) = [y^2*z^9]' + x^2*[y^27*z^3]' = [y*z]'*exp(x*y*z)
#f_y(z(x,y)) = (2*y*z^9 + y^2*9*z^8*z') + x^2*(27*y^26*z^3 + y^27*3*z^2*z')
#            = (z+y*z')*exp(x*y*z)
#f_y(z(x,y)) = 2*y*z^9 + 9*y^2*z^8*z' + 27*x^2*y^26*z^3 + 3*x^2*y^27*z^2*z'
#	     = z*exp(x*y*z) + y*z'*exp(x*y*z)
#f_y(z(x,y)) = 9*y^2*z^8*z' + 3*x^2*y^27*z^2*z' - y*z'*exp(x*y*z)
#	     = z*exp(x*y*z) - 2*y*z^9 - 27*x^2*y^26*z^3
#f_y(z(x,y)) = z'*(9*y^2*z^8 + 3*x^2*y^27*z^2 - y*exp(x*y*z))
#	     = z*exp(x*y*z) - 2*y*z^9 - 27*x^2*y^26*z^3
#f_y(z(x,y)) = z' = (z*exp(x*y*z) - 2*y*z^9 - 27*x^2*y^26*z^3) 
#		    / (9*y^2*z^8 + 3*x^2*y^27*z^2 - y*exp(x*y*z))



#ATTENDANCE QUESTION 3:
#DO THE SECOND PART, FIND f_y(1,1)

#ANSWER TO QUESTION 3:
#f_y(x,y) = (x^2+y^2) = (x*y*z)' + (x*y*z^5)'
#f_y(x,y) = 2*y = x*(y*z)' + x*(y*z^5)'
#f_y(x,y) = 2*y = x*(z+y*z') + x*(z^5 + y*4*z^5*z') 
#f_y(x,y) = 2*y = x*z + x*y*z' + x*z^5 + x*y*4*z^5*z'
#z'(1,1) = 2*1 = 1*1 + 1*1*z' + 1*1^5 + 1*1*4*1^5*z'
#z'(1,1) = 2 = 1 + z' + 1 + 4*z'
#z'(1,1) = 2 = 2 + 5*z'
#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 TO QUESTION 4:
#The answer is the same. Since x and y (and z) 
#coordinates are equal, and the x and y aspects of
#the function have the same qualities (on one side, both x and y are
#squared, on the other side, x and y are multiplied together
#and both have the same degree), the answers should be the same.



#ATTENDANCE QUESTION 5:
#Let a and b BE AS ABOVE (0 and 9)
#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 TO QUESTION 5:
#z = x^0 + y^9 + 0*9*x^y AT THE POINT (1,1,2+0*9)
#z = 1 + y^9 AT THE POINT (1,1,2)
#1 = 1 + 1^9 = TRUE
#f_x(x,y) = 1
#f_x(1,1) = 1
#f_y(x,y) = 9*y^8
#f_y(1,1) = 9*1^8 = 9
#z-2 = (x-1) + 9*(y-1)
#z = x - 1 + 9*y - 9 + 2
#z = x + 9*y - 8