Do 3 4 10 and 1 5 30 He on the same level surface of the fun

Do (3, 4, 10) and (1, 5, 30) He on the same level surface of the function f (x, y, z) = z- x^2 - y^2 Explain. Describe the level cm ves of the function f (x, y) = y - cos (x)

Solution

11.

We substitute the points and check . If they both give the same function value then yes they lie on the same level surface

f(3,4,10)=10-3^2-4^2=10-25=-15

f(1,5,30)=30-1^2-5^2=4

Hence they do not lie on the same level surface

12

Level curves of this function are

f(x,y)=C for some constant C, for each real number C we have one level curve

y-cos(x)=C

y=cos(x)+C

So we have a sinusoidal periodic function with period 2pi shifted up by C units.