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

$$\nabla f(1,2) \cdot \frac{\vec{v}}{\|\vec{v}\|}$$

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 the xy$-plane, using polar coordinates?

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

Q: Which of the following is the correct formula for the curvature $$\kappa of a $$smooth curve given by $$\vec{r}(t)$$?

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

Q: Which of the following is the correct setup for the volume of the region inside the cylinder $$x^2$$ + $$y^2 = 4$ and between z = 0$ and z = x + y$ using cylindrical coordinates?

$$\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{r \cos \theta + r \sin \theta} r \, dz \, dr \, d\theta$$

Question 1

Which of the following is the correct formula for the area enclosed by one loop of the polar curve $$r = 2 \sin(3	heta)$$?

Accepted Answer

$$A = \frac{1}{2} \int_{0}^{\pi/3} (2 \sin(3\$$theta))^2$$ \, d	heta$$

Question 2

A particle moves in space with position vector $$\vec{r}(t) = \langle t^2$, \sin t, e^t \rangle. $$What is the velocity vector at $$t = 0$$?

Accepted Answer

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

Question 3

Which of the following statements about the cross product of two vectors $$\vec{a}$$ and $$\vec{b} in$$ $$\mathbb{R}^3 is $$TRUE?

Accepted Answer

The cross product is a vector perpendicular to both $$\vec{a}$$ and $$\vec{b}.$$

Question 4

Given the plane $$2x - 3y + 4z = 12$$, which of the following is a normal vector to the plane?

Accepted Answer

$$\langle 2, -3, 4 \rangle$$

Question 5

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

Accepted Answer

$$
abla f(1,2) \cdot \frac{\vec{v}}{\|\vec{v}\|}$$

Question 6

Evaluate the double integral $$\iint_R x y \, dA$$ where $$R is $$the rectangle $$0 \leq x \leq 2$$, $$0 \leq y \leq 3.$$

Accepted Answer

$$12$$

Question 7

Which of the following best describes the gradient vector $$
abla f(x, y) at a $$point $$(x_0$, y_0$)$$?

Accepted Answer

It points in the direction of greatest increase of $$f at$$ $$(x_0$, y_0$).$$

Question 8

Find the equation of the tangent plane to the surface $$z = x^2$ + y^2$ at the point (1$$, 2, 5)$$.

Accepted Answer

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

Question 9

A company wants to maximize its profit $$P(x, y) = 100x + 150y$$ subject to the constraint $$2x + 3y = 60. $$Which method should be used to solve this optimization problem?

Accepted Answer

Lagrange multipliers

Question 10

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 the xy$-plane, using polar coordinates?

Accepted Answer

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

Question 11

If $$\vec{a} = \langle 1, 2, 3 \rangle$$ and $$\vec{b} = \langle 4, 5, 6 \rangle$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$32$$

Question 12

Which of the following is the correct formula for the curvature $$\kappa of a $$smooth curve given by $$\vec{r}(t)$$?

Accepted Answer

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

Question 13

Evaluate the triple integral $$\iiint_E z \, dV$$ where $$E is $$the region bounded by $$z = 0$$ and $$z = 1 - x^2$ - y^2$ in the xy$-plane.

Accepted Answer

$$\frac{\pi}{8}$$

Question 14

Which of the following is the correct statement 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 15

A curve is given by the parametric equations $$x = t^2$, y$$ = $$t^3. $$What is $$\frac{dy}{dx} at$$ $$t = 2$$?

Accepted Answer

$$3t/2$$

Question 16

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

Accepted Answer

$$\int_{a}^{b} \|\vec{r}'(t)\| \, dt$$

Question 17

Suppose $$f(x, y) = x^2$ + y^2$. Which of the following points is a critical point?

Accepted Answer

$$(0, 0)$$

Question 18

Which of the following is the correct setup for the volume of the region inside the cylinder $$x^2$$ + $$y^2 = 4$ and between z = 0$ and z = x + y$ using cylindrical coordinates?

Accepted Answer

$$\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{r \cos 	heta + r \sin 	heta} r \, dz \, dr \, d	heta$$

Question 19

Which of the following best describes a level curve of a function $$f(x, y)$$?

Accepted Answer

A curve along which $$f(x, y) is $$constant.

Calculus III Syllabus Overview and Key Topics

Guida di studio - Domande di pratica

Study guide - Flashcard