Solutions to Problem Set 8

Math 311-01/02

2-6-2018

1. Determine whether the given point lies on the surface $z = x^3 + 2 y^2$ .

(a) $(2, 3, 26)$ .

(b) $(-1, 1, 1)$ .

(c) $(-2, 1, 10)$ .

In each case, plug the x and y-coordinates into the surface equation and see if the result is equal to the z-coordinate. If it is, the point lies on the surface; if it isn't, the point does not lie on the surface.

(a)

$$2^3 + 2 \cdot 3^2 = 8 + 18 = 26.$$

The point lies on the surface.

(b)

$$(-1)^3 + 2 \cdot 1^2 = -1 + 2 = 1.$$

The point lies on the surface.

(c)

$$(-2)^3 + 2 \cdot 1^2 = -8 + 2 = -6 \ne 10.$$

The point does not lie on the surface.


2. Sketch the surface $z = 4 x^2 +
   4 y^2$ .

$$\hbox{\epsfysize=1.5in \epsffile{surfaces-problems-1.eps}}\quad\halmos$$


3. Sketch the surface $z = 3 -
   x^2 - y^2$ .

$$\hbox{\epsfysize=1.5in \epsffile{surfaces-problems-2.eps}}\quad\halmos$$


4. Sketch the surface $z = x^2 -
   y^2 + 2$ .

$$\hbox{\epsfysize=1.5in \epsffile{surfaces-problems-3.eps}}\quad\halmos$$


5. Sketch the surface $z = x^2 +
   3$ .

$$\hbox{\epsfysize=1.5in \epsffile{surfaces-problems-4.eps}}\quad\halmos$$


6. Sketch the surface $z = \sin
   x$ .

$$\hbox{\epsfysize=1.5in \epsffile{surfaces-problems-5.eps}}\quad\halmos$$


7. Sketch the surface $z = (x^2 +
   4 y^2) e^{(-x^2 - y^2)}$ .

$$\hbox{\epsfysize=1.5in \epsffile{surfaces-problems-6.eps}}\quad\halmos$$


8. Sketch the surface $z =
   \sqrt{x^2 + y^2} + 2$ .

$$\hbox{\epsfysize=1.5in \epsffile{surfaces-problems-7.eps}}\quad\halmos$$


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