#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

1. Question: Let a be the 5th digit of your RUID and let be 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(x,y). 

Answer:

f(x,y)=1+9y^3

df/dx=0
df/dy=27y^2

2. Question: With a and b above, find f_x (z(x,y)) and f_y(z(x,y)) if x,y,z are related by the relationship
x^a * y^2 * z^b + x^2*y^(3*b)*z^3=exp(x*y*z)

Answer:

f_x=9z^8+2xz^3+x^2z'z^2
f_y=2yz^9+9z'z^8y^2+27y^26z^3+3z^2z'y^27

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

x^2 + y^2 = x*y*z +x*y*z^5

dz/dy = 2y = x*(y*z)' + x*(y*z^5)
z' = 2y = x*(z + yz') + x*(5z^4y + z^5)
z' at (1,1) = 2 = 1 + z' + 5z' + 1
z' = 0

f_y(1,1) = 0

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

Yes, the answer is the same and you could have predicted it without the calculation since
both variables, x and y, have the "relationship" to z in the equation. 

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

Answer: 

a = 0
b = 9
z_0 = 1 + 1 = 2 =2 works

f_x = 0
f_y = 9y^8
f_x(1,1) = 0
f_y(1,1) = 9
z-2 = 0*(x-1) + 9*(y-1)
z = 9y - 7