Q: Which of the following is the correct representation of the area enclosed by one loop of the polar curve $$r = 2\sin\theta$$?

$$A = \int_{0}^{\pi} \frac{1}{2} (2\sin\$$theta)^2$$ d\theta$$

Q: Given vectors $$\vec{a} = \langle 1, 2, 3 \rangle$$ and $$\vec{b} = \langle 4, 0, -1 \rangle$$, what is their dot product?

$$1 \times 4 + 2 \times 0 + 3 \times (-1) = 1$$

Q: Given $$f(x, y) = x^2$y + y^3$, what is the value of the directional derivative at the point (1$$,2)$$ in the direction of the vector $$\vec{v} = \langle 3, 4 \rangle$$?

$$\nabla f(1,2) \cdot \frac{\vec{v}}{\|\vec{v}\|} = \langle 4, 13 \rangle \cdot \langle 0.6, 0.8 \rangle = 4 \times 0.6 + 13 \times 0.8 = 1.6 + 10.4 = 12$$

Q: Which of the following is the correct setup for the volume of the solid bounded above by $$z = 4 - x^2$ - y^2$ and below by z = 0$ using polar coordinates?

$$\int_{0}^{2\pi} \int_{0}^{2} (4 - r^2$) r \, dr \, d\theta$$

Question 1

A particle moves along the curve defined by the parametric equations $$x = 2t^2$, y$$ = $$3t^3. $$What is the slope of the tangent to the curve at $$t = 1$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{9}{4}$$

Question 2

Which of the following is the correct representation of the area enclosed by one loop of the polar curve $$r = 2\sin	heta$$?

Accepted Answer

$$A = \int_{0}^{\pi} \frac{1}{2} (2\sin\$$theta)^2$$ d	heta$$

Question 3

Given vectors $$\vec{a} = \langle 1, 2, 3 \rangle$$ and $$\vec{b} = \langle 4, 0, -1 \rangle$$, what is their dot product?

Accepted Answer

$$1 	imes 4 + 2 	imes 0 + 3 	imes (-1) = 1$$

Question 4

Which of the following best describes the geometric meaning of the cross product $$\vec{a} 	imes \vec{b} of $$two vectors in $$\mathbb{R}^3$$?

Accepted Answer

A vector perpendicular to both $$\vec{a}$$ and $$\vec{b}$$ with magnitude equal to the area of the parallelogram they span.

Question 5

Find the equation of the plane passing through the point $$(1,2,3)$$ and perpendicular to the vector $$\vec{n} = \langle 2, -1, 4 \rangle.$$

Accepted Answer

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

Question 6

A particle moves in space with position vector $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle. $$What is the speed of the particle at $$t = 2$$?

Accepted Answer

$$\sqrt{1 + 16 + 144} = \sqrt{161}$$

Question 7

Given $$f(x, y) = x^2$y + y^3$, what is the value of the directional derivative at the point (1$$,2)$$ in the direction of the vector $$\vec{v} = \langle 3, 4 \rangle$$?

Accepted Answer

$$
abla f(1,2) \cdot \frac{\vec{v}}{\|\vec{v}\|} = \langle 4, 13 angle \cdot \langle 0.6, 0.8 angle = 4 	imes 0.6 + 13 	imes 0.8 = 1.6 + 10.4 = 12$$

Question 8

Which of the following statements about the gradient vector $$
abla f(x, y) is $$true?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

A company wants to maximize its profit $$P(x, y) = 100x + 150y - x^2$ - y^2$ subject to the constraint x + y = 10$. Using Lagrange multipliers, what is the value of x$ at the maximum?

Accepted Answer

$$x = 4$$

Question 11

Evaluate the double integral $$\iint_{R} x y \, dA$$ over the rectangle $$R: 0 \leq x \leq 2, 0 \leq y \leq 3.$$

Accepted Answer

$$\int_{0}^{2} \int_{0}^{3} x y \, dy \, dx = 9$$

Question 12

Which of the following is the correct setup for the volume of the solid bounded above by $$z = 4 - x^2$ - y^2$ and below by z = 0$ using polar coordinates?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{2} (4 - r^2$) r \, dr \, d	heta$$

Question 13

Find the length of the curve $$y = x$$^{3/2}$$ from x = 0$ to x = 4$.

Accepted Answer

$$\int_{0}^{4} \sqrt{1 + \left(\frac{3}{2} x$$^{1/2}$$\$$right)^2$$} \, dx$$

Question 14

A hospital is monitoring the spread of a contaminant in a circular pond of radius 10 meters. The concentration at a point $$(r, 	heta) is $$given by $$C(r, 	heta) = 100 - r^2$. What is the total amount of contaminant in the pond?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{10} (100 - r^2$) r \, dr \, d	heta$$

Question 15

Which of the following is the correct expression for the curvature $$\kappa of $$the curve $$\vec{r}(t)$$?

Accepted Answer

$$\kappa = \frac{\|\vec{r}'(t) 	imes \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}$$

Question 16

Evaluate the triple integral $$\iiint_{E} z \, dV$$ where $$E is $$the region bounded by $$z = 0$$, $$z = 1 - x^2$ - y^2$, and x^2$ + y^2$ \leq 1.$$

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{1 - r^2$} z r \, dz \, dr \, d	heta$$

Question 17

Which of the following is the correct form of the chain rule for $$z = f(x, y)$$ where $$x = g(t)$$ and $$y = h(t)$$?

Accepted Answer

$$\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$$

Question 18

A pharmaceutical company is testing a new drug. The concentration of the drug in the bloodstream at time $$t (in $$hours) is given by $$C(t) = 5e$$^{-0.2t}$$. What is the total amount of drug in the bloodstream from t = 0$ to t = 10$ hours?

Accepted Answer

$$\int_{0}^{10} 5e$$^{-0.2t}$$ dt$$

Question 19

Which of the following is the correct linear approximation of $$f(x, y) = e$$^{xy}$$ at the point (0$$, 0)$$?

Accepted Answer

$$f(x, y) \approx 1 + yx$$

Calculus III Syllabus: Multivariable Calculus and Vector Analysis

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