$$2 \times 1 + (-1) \times 4 + 3 \times (-2) = 2 - 4 - 6 = -8$$

$$\nabla f = \lambda \nabla g$$

Question 1

Which of the following best describes a vector in three-dimensional space?

Accepted Answer

A quantity with both magnitude and direction.

Question 2

Given vectors $$\mathbf{a} = \langle 2, -1, 3 \rangle$$ and $$\mathbf{b} = \langle 1, 4, -2 \rangle$$, what is their dot product?

Accepted Answer

$$2 	imes 1 + (-1) 	imes 4 + 3 	imes (-2) = 2 - 4 - 6 = -8$$

Question 3

Which equation represents a plane in space passing through the point $$(1,2,3)$$ with normal vector $$\mathbf{n} = \langle 2, -1, 4 \rangle$$?

Accepted Answer

$$2(x-1) - 1(y-2) + 4(z-3) = 0$$

Question 4

A particle moves along the curve $$\mathbf{r}(t) = \langle t^2$, \sin t, e^t \rangle. $$What is the velocity vector at $$t=0$$?

Accepted Answer

$$\langle 0, 1, 1 \rangle$$

Question 5

Which of the following is the correct formula for the length of a curve $$\mathbf{r}(t)$$ from $$t=a to$$ $$t=b$$?

Accepted Answer

$$\int_a^b \| \mathbf{r}'(t) \| \, dt$$

Question 6

If $$f(x,y) = x^2$y + y^3$, what is the partial derivative $$\frac{\partial f}{\partial y}$$?

Accepted Answer

$$x^2$$ + $$3y^2$$

Question 7

Which statement about the gradient vector $$
abla f is $$correct?

Accepted Answer

It points in the direction of greatest increase of $$f.$$

Question 8

Suppose $$f(x,y) = x^2$ + y^2$. What is the equation of the tangent plane at the point (1$$,2,5)$$?

Accepted Answer

$$z = 2(x-1) + 4(y-2) + 5$$

Question 9

A function $$f(x,y)$$ has a critical point at $$(2,3). $$Which method can be used to determine if this point is a maximum, minimum, or saddle point?

Accepted Answer

Second derivative test using the Hessian matrix.

Question 10

Which formula is used to solve constrained optimization problems using Lagrange multipliers?

Accepted Answer

$$
abla f = \lambda 
abla g$$

Question 11

Evaluate the double integral $$\$$int_0$$^1 \$$int_0$$^2 (x+y) \, dy \, dx.$$

Accepted Answer

$$3$$

Question 12

Which coordinate system is most appropriate for integrating over a circular region?

Accepted Answer

Polar coordinates.

Question 13

The triple integral $$\int \int \int_E \rho(x,y,z) \, dV is $$used to calculate which physical quantity?

Accepted Answer

Mass of a solid with density $$\rho(x,y,z).$$

Question 14

Which theorem relates a line integral around a simple closed curve to a double integral over the region it encloses?

Accepted Answer

Green's Theorem.

Question 15

If $$\mathbf{F} = \langle y, -x \rangle$$, what is the curl of $$\mathbf{F} in $$two dimensions?

Accepted Answer

$$-2$$

Question 16

Which of the following is a necessary condition for a vector field $$\mathbf{F} to be $$conservative?

Accepted Answer

The curl of $$\mathbf{F} is $$zero.

Question 17

Stokes' Theorem relates which two types of integrals?

Accepted Answer

A surface integral of the curl of a vector field and a line integral around the boundary.

Question 18

In electromagnetism, the Divergence Theorem is used to relate which two quantities?

Accepted Answer

The flux of a vector field through a closed surface and the divergence over the volume inside.

Question 19

A vector field $$\mathbf{F} is $$given by $$\mathbf{F}(x,y,z) = \langle x, y, z \rangle. $$What is the divergence of $$\mathbf{F}$$?

Accepted Answer

$$1 + 1 + 1 = 3$$

Calculus III (Multivariable Calculus) - Course Overview and Study Guide

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