Question is
Let f(x, y) = 100 + 10x + 25y – x^2 – 5y^2.
a. Describe the cross section of the surface Z = f(x, y) produced by cutting it with the planes Y = 0, y = 1, y = 2, and y = 3.
b. Describe the cross section of the surface in the planes x = 0, x = 1, x = 2, and x = 3.
c. Describe the surface z = f(x, y)
What I have done for part b:
Z = F(0,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(0) + 25y – (0)^2 – 5y^2
= 100 + 25y - 5y^2
z = F(1,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(1) + 25y – (1)^2 – 5y^2
= 111 + 25y - 5y^2
z = F(2,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(2) + 25y – (2)^2 – 5y^2
= 124 + 25y - 5y^2
z = F(3,y) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(3) + 25y – (3)^2 – 5y^2
= 139 + 25y - 5y^2
How i described the results:
The cross section of f(x, y) = 100 + 10x + 25y – x2 – 5y2 that correspond to x = 0, x = 1, x = 2, and x = 3 are upper semicircles with centres (0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0) respectively.
Is this correct so far?
For part A i did the following and corrcted my explanation
Z = F(x,0) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(0) – x^2 – 5(0)^2
= -x^2 + 10x + 100z
F(x,1) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(1) – x^2 – 5(1)^2
= - x^2 + 10x + 120
z = F(x,2) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(2) – x^2 – 5(2)^2
= - x^2 + 10x + 130
z = F(x,3) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10x + 25(3) – x^2 – 5(3)^2
= - x^2 + 10x + 130
The cross section of f(x, y) = 100 + 10x + 25y – x^2 – 5y^2 produced by cutting it with the plane y = 0, y = 1, y = 2, and y = 3 shows that each of these is a parabola that opens downward.
What I have done so far for part C
Describe the surface z = f(x, y)
z = f(0, 0) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(0) + 25(0) – (0)^2 – 5(0)^2
= 100
z = f(1, 1) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(1) + 25(1) – (1)^2 – 5(1)^2
= 141
z = f(2, 2) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(2) + 25(2) – (2)^2 – 5(2)^2
= 194
z = f(3, 3) = 100 + 10x + 25y – x^2 – 5y^2
= 100 + 10(3) + 25(3) – (3)^2 – 5(3)^2
= 259
My textbook has not really explained to me what I had to do for part C so I assumed I did the above correctly. I am not sure how to explain my results.