Question 1

The first image shows a 3D surface with a bowl-like shape and striped pattern. Which of the following equations best describes the surface shown in the image?

Accepted Answer

$$z = x^2$ + y^2$

Question 2

The second image shows a 3D object with a circular cross-section and a striped pattern, intersected by three colored axes. If the red axis represents the $$x-$$axis, the blue axis the $$y-$$axis, and the black axis the $$z-$$axis, which type of surface is most likely depicted?

Accepted Answer

A paraboloid

Question 3

Suppose the surface in the first image is given by $$z = x^2$ + y^2$ for x^2$ + y^2$ \leq R^2$. What is the volume enclosed by this surface and the plane z = 0$?

Accepted Answer

$$V = \frac{1}{2} \pi R^4$

Question 4

If the striped bowl in the first image is rotated about the $$z-$$axis, which method would you use to compute its volume?

Accepted Answer

Disk method

Question 5

Given the surface $$z = x^2$ + y^2$, what is the equation for the cross-section of the surface at z = c$?

Accepted Answer

$$x^2$$ + $$y^2 = c$

Question 6

If the bowl-shaped surface in the first image is described by $$z = x^2$ + y^2$, what is the gradient vector $$
abla z$$ at the point (x$$, y)$$?

Accepted Answer

$$
abla z = (2x, 2y)$$

Question 7

For the surface $$z = x^2$ + y^2$, what is the value of the second partial derivative $$\frac{\$$partial^2 z$$}{\partial $$x^2$$}$$?

Accepted Answer

$$2$$

Question 8

If you were to compute the surface area of the bowl-shaped surface $$z = x^2$ + y^2$ over the disk x^2$ + y^2$ \leq R^2$, which integral would you set up?

Accepted Answer

$$\$$int_0$$^{2\pi} \$$int_0$$^R \sqrt{1 + 4r^2$} \, r \, dr \, d	heta$$

Question 9

The second image shows a cross-section of a surface. If the surface is $$z = x^2$ + y^2$, what is the shape of the cross-section when sliced by the plane y = 0$?

Accepted Answer

A parabola

Question 10

If the bowl-shaped surface in the first image is used to model a physical object, which property is maximized at the center $$(0,0)$$?

Accepted Answer

Depth

Volumes of Solids of Revolution: The Paraboloid Example

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