Assessment Quiz: Lesson 1: Three-Dimensional Cartesian Coordinates and Vector Operations

Question 1

If vector A = (2, -1, 3) and vector B = (k, 4, -2) are perpendicular, what is the value of k?

Accepted Answer

5

Answer

-5

Answer

10

Answer

-10

Question 2

If $y = \sec(3x)$, then the derivative $\frac{dy}{dx}$ is equal to:

Accepted Answer

A) $3\sec(3x) 	an(3x)$

Answer

B) $\sec(3x) 	an(3x)$

Answer

C) $-3\csc(3x) \cot(3x)$

Answer

D) $3\sec^2(3x)$

Question 3

The derivative of $f(x) = 5^{x^2}$ with respect to $x$ is:

Accepted Answer

A) $2x \cdot 5^{x^2}$

Answer

B) $2x \cdot 5^{x^2} \cdot \ln 5$

Answer

C) $5^{x^2} / \ln 5$

Answer

D) $x^2 \cdot 5^{x^2 - 1}$

Question 4

If $x = \sin t$ and $y = \cos t$, then the second derivative $\frac{d^2y}{dx^2}$ is equal to:

Accepted Answer

A) $-\cos t$

Answer

B) $-\sec^3 t$

Answer

C) $-	an t$

Answer

D) $\csc^2 t$

Question 5

The slope of the normal to the curve $y = e^{2x}$ at the point where $x = 0$ is:

Accepted Answer

A) 2

Answer

B) -2

Answer

C) 1/2

Answer

D) -1/2

Question 6

The n-th term of an arithmetic sequence with first term a and common difference d is given by:

Accepted Answer

T_n = a + (n - 1) * d

Answer

T_n = a * r^(n - 1)

Answer

T_n = a + n * d

Answer

T_n = 1/2 n (2a + d)

Question 7

In the arithmetic sequence (3, 7, 11, 15, ...), the common difference d and the 20th term (T_20) are respectively:

Accepted Answer

d = 4 and T_20 = 79

Answer

d = 4 and T_20 = 83

Answer

d = 3 and T_20 = 60

Answer

d = 5 and T_20 = 98

Question 8

The sum of the first n terms of an arithmetic sequence (S_n) is formulated as:

Accepted Answer

S_n = n / 2 * [ 2a + (n - 1) * d ] = n / 2 * [ a + l ]

Answer

S_n = a * (1 - r^n) / (1 - r)

Answer

S_n = n * (a + d)

Answer

S_n = a / (1 - r)

Question 9

The common ratio r of the geometric sequence (64, 32, 16, 8, ...) is:

Accepted Answer

r = 1/2

Answer

r = 2

Answer

r = -1/2

Answer

r = 4

Question 10

The sum to infinity (S_infinity) of a geometric series exists and converges if and only if the common ratio satisfies:

Accepted Answer

|r| < 1 (i.e. -1 < r < 1), and S_infinity = a / (1 - r)

Answer

r > 1

Answer

r = 1

Answer

|r| >= 2

Point of comparison	Force depends on time $t$	Force depends on position $s$	Force depends on velocity $v$
Equation used	$m \frac{d v}{d t} = F (t)$	$m v \frac{d v}{d s} = F (s)$	$m \frac{d v}{d t} = F (v)$ or $m v \frac{d v}{d s} = F (v)$
Variables separated	$d v = \frac{F (t)}{m} d t$	$v d v = \frac{F (s)}{m} d s$	$\frac{d v}{F (v)} = \frac{d t}{m}$ or $\frac{v d v}{F (v)} = \frac{d s}{m}$
First answer	$v (t)$	$v (s)$ (speed at a place)	$v (t)$ or $v (s)$
Then	integrate again for $s (t)$	$\frac{d s}{d t} = v (s)$ for $t$	integrate for the other variable

2-1 (a) Newton's second law as a differential equation

01 2-1 (a) Newton's second law as a differential equation

The law

Three forms of the acceleration

Integrals that follow

General method

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