Question 1

Which of the following is the correct definition of the hyperbolic sine function $$\sinh(x)$$?

Accepted Answer

$$\sinh(x) = \dfrac{e^x - e$$^{-x}$$}{2}$$

Question 2

Which of the following series converges?

Accepted Answer

$$\sum\limits_{n=1}^{\infty} \dfrac{1}{n^2$}$$

Question 3

Which of the following is the Maclaurin series for $$\cos(x)$$?

Accepted Answer

$$\sum\limits_{n=0}^{\infty} \dfrac{(-1)^n x$$^{2n}$$}{(2n)!}$$

Question 4

Suppose $$f(x) = e^x. $$Using Taylor's theorem, what is the remainder $$R_2(x$$)$$ when approximating f(x$$)$$ by its second-degree Taylor polynomial centered at x=0$?

Accepted Answer

$$R_2(x$$) = \dfrac{e^c}{3!}$$x^3$$ for some $$c$$ between $$0$$ and $$x$$

Question 5

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

Accepted Answer

$$\langle 1, 4, 12 \rangle$$

Question 6

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

Accepted Answer

$$-8$$

Question 7

Which of the following is the equation of the plane passing through the point $$(1,2,3)$$ and normal to the vector $$\langle 2, -1, 4 \rangle$$?

Accepted Answer

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

Question 8

Which of the following is the equation of a right circular cylinder with axis along the $$z-$$axis and radius $$3$$?

Accepted Answer

$$x^2$$ + $$y^2 = 9$

Question 9

Let $$f(x, y) = x^2$ + y^2$. What are the level curves of f$?

Accepted Answer

Circles centered at the origin

Question 10

Given $$f(x, y) = x^2$y + y^3$, what is $$\dfrac{\partial f}{\partial y}$$?

Accepted Answer

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

Question 11

Suppose $$z = f(x, y)$$ where $$x = t^2$ and y$$ = \sin t$$. Using the chain rule, what is $$\dfrac{dz}{dt}$$?

Accepted Answer

$$\dfrac{\partial f}{\partial x} \cdot 2t + \dfrac{\partial f}{\partial y} \cdot \cos t$$

Question 12

The gradient of $$f(x, y) = x^2$ + y^2$ at the point (1$$, -2)$$ is:

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

$$(0, 0)$$

Question 15

Suppose you want to maximize $$f(x, y) = xy$$ subject to the constraint $$x^2$$ + $$y^2 = 1$. Which method should you use?

Accepted Answer

Lagrange multipliers

Question 16

A force $$\vec{F} = \langle 3, 4, 0 \rangle$$ moves an object from $$(0,0,0) to$$ $$(2,1,0). $$What is the work done by the force?

Accepted Answer

$$10$$

Question 17

Which of the following conic sections is represented by the equation $$x^2$$ - $$y^2 = 1$?

Accepted Answer

Hyperbola

Question 18

Which of the following is the correct formula for the cross product $$\vec{a} 	imes \vec{b}$$ for $$\vec{a} = \langle a_1$, a_2$, a_3$ angle$$ and $$\vec{b} = \langle b_1$, b_2$, b_3$ angle$$?

Accepted Answer

$$\langle a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1 \rangle$$

Calculus Study Guide: Advanced Topics and Multivariable Calculus

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