Q: Let $$A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 0 \\ 4 & 2 \end{bmatrix}. $$Compute $$AB.$$

$$\begin{bmatrix} 6 & 2 \\ 4 & 6 \end{bmatrix}$$

Q: Which of the following statements about the cross product of two nonzero vectors $$\vec{v}$$ and $$\vec{w} in$$ $$\mathbb{R}^3 is $$TRUE?

$$\vec{v} \times \vec{w} is $$always perpendicular to both $$\vec{v}$$ and $$\vec{w}$$

Question 1

Let $$A = \begin{bmatrix} 2 & 1 \ 0 & 3 \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 0 \ 4 & 2 \end{bmatrix}. $$Compute $$AB.$$

Accepted Answer

$$\begin{bmatrix} 6 & 2 \ 4 & 6 \end{bmatrix}$$

Question 2

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

Accepted Answer

$$\vec{v} 	imes \vec{w} is $$always perpendicular to both $$\vec{v}$$ and $$\vec{w}$$

Question 3

Suppose $$f(x) = x^3$ - 3x on$$ $$[-3,3]. $$What is the global maximum value of $$f on $$this interval?

Accepted Answer

$$f(3) = 18$$

Question 4

Evaluate $$\displaystyle \$$int_0$$^1 6x^2$ dx.$$

Accepted Answer

$$2$$

Question 5

Which of the following is the Taylor series for $$e^x$$ centered at $$a=0$$?

Accepted Answer

$$\displaystyle \sum_{k=0}^\infty \frac{x^k}{k!}$$

Question 6

A solid is formed by rotating the region bounded by $$y = x^2$ and y = 0$ from x=0$ to x=2$ about the x$-axis. What is the volume of the solid?

Accepted Answer

$$\displaystyle \$$int_0$$^2 \pi (x^2$)^2 dx$

Question 7

Which of the following is the correct formula for the derivative of $$\arcsinh x$$?

Accepted Answer

$$\displaystyle \frac{1}{\sqrt{1 + x^2$}}$$

Question 8

Let $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}. $$What is $$\det(A)$$?

Accepted Answer

$$-2$$

Question 9

Suppose $$f(x) is $$continuous on $$[a,b]$$ and $$F(x) is an $$antiderivative of $$f(x). $$According to the Fundamental Theorem of Calculus, $$\int_a^b f(x) dx$$ equals:

Accepted Answer

$$F(b) - F(a)$$

Question 10

Which of the following integrals can be evaluated using the substitution $$x = a \sin 	heta$$?

Accepted Answer

$$\displaystyle \int \sqrt{a^2$ - x^2$} dx$$

Question 11

A company produces widgets. The cost to produce $$x$$ widgets is $$C(x) = 100 + 5x. $$What is the marginal cost at $$x=10$$?

Accepted Answer

$$5$$

Question 12

Let $$\vec{v} = \langle 1, 2, 2 \rangle. $$What is the length (norm) of $$\vec{v}$$?

Accepted Answer

$$3$$

Question 13

Which of the following is the correct partial fraction decomposition for $$\displaystyle \frac{3x+5}{(x+1)(x-2)}$$?

Accepted Answer

$$\displaystyle \frac{A}{x+1} + \frac{B}{x-2}$$

Question 14

Evaluate $$\displaystyle \$$int_0$$^{\pi} \sin x dx.$$

Accepted Answer

$$2$$

Question 15

Which of the following is the correct formula for the area of the surface generated by revolving $$y = f(x)$$, $$a \leq x \leq b$$, about the $$x-$$axis?

Accepted Answer

$$\displaystyle \int_a^b 2\pi f(x) \sqrt{1 + [f'(x)]^2} dx$$

Question 16

Let $$n be a $$natural number. According to the binomial theorem, the coefficient of $$x^2 in$$ $$(1 + x)^5$ is:

Accepted Answer

$$10$$

Question 17

Which of the following is the correct formula for the arc length of the curve $$y = f(x)$$ from $$x=a to$$ $$x=b$$?

Accepted Answer

$$\displaystyle \int_a^b \sqrt{1 + [f'(x)]^2} dx$$

Question 18

Suppose $$\vec{u} = \langle 2, -1, 3 \rangle$$ and $$\vec{v} = \langle 1, 4, -2 \rangle. $$What is $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$-8$$

Question 19

Which of the following is the correct formula for the number of ways to choose $$r$$ objects from $$n$$ distinct objects (order does not matter)?

Accepted Answer

$$\displaystyle \frac{n!}{r!(n-r)!}$$

MTHS121: Introductory Algebra and Calculus II – Structured Study Notes

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