Question 1

The first image shows a 3D surface with a bowl-like shape and vertical stripes. Which of the following equations most likely describes the surface shown in the image?

Accepted Answer

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

Question 2

In the second image, the surface is shown with three colored axes. If the red axis represents the $$x-$$axis, the blue axis represents the $$y-$$axis, and the vertical axis represents the $$z-$$axis, which of the following best describes the orientation of the surface?

Accepted Answer

The surface is symmetric about the $$z-$$axis.

Question 3

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

Accepted Answer

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

Question 4

If the surface shown is rotated about the $$z-$$axis, what type of solid is generated?

Accepted Answer

A paraboloid of revolution

Question 5

Given the surface $$z = x^2$ + y^2$, what is the area of the surface above the disk x^2$ + y^2$ \leq R^2$?

Accepted Answer

$$A = 2\pi \$$int_0$$^R r \sqrt{1 + 4r^2$} dr$$

Question 6

If the surface in the first image is described by $$z = x^2$ + y^2$, what is the gradient $$
abla z$$ at the point (1$$,1)$$?

Accepted Answer

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

Question 7

What is the equation of the tangent plane to the surface $$z = x^2$ + y^2$ at the point (a$$, b)$$?

Accepted Answer

$$z = 2a(x-a) + 2b(y-b) + a^2$ + b^2$

Question 8

If the surface is $$z = x^2$ + y^2$, what is the minimum value of z$ on the disk x^2$ + y^2$ \leq R^2$?

Accepted Answer

$$z = 0$$

Question 9

If the surface in the first image is $$z = x^2$ + y^2$, what is the value of the double integral $$\iint_{$$x^2$$ + $$y^2$$ \leq 1} z \, dx \, dy$$?

Accepted Answer

$$\frac{\pi}{2}$$

Question 10

Which of the following best describes the cross-section of the surface $$z = x^2$ + y^2$ at a fixed value of z = c$?

Accepted Answer

A circle of radius $$\sqrt{c}$$

Calculus I: Functions, Limits, Derivatives, and Integrals – Comprehensive Study Notes

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