Vector Operations and Magnitude

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Questions and Answers

If Higgins invests $2500, how much will she have after 7.5 years if the interest rate is 14.23% compounded monthly?

6250

5000

7500 (correct)

3750

What is the value of log a 6 in terms of x and y, given log a 2 = x and log a 3 = y?

y - x

x - y

x + y (correct)

x / y

What is the domain of f(x) = -log2(x + 1)?

(-∞, 1)

(-∞, -1)

(-1, ∞) (correct)

(-∞, ∞)

What is the equation of the graph of f(x) = 3x shifted 4 units to the right?

f(x) = 3(x - 4) Signup and view all the answers

Using an exponential model, what is the number of travelers (in millions) to foreign countries in 2005?

66.1 Signup and view all the answers

If a tower that is 125 feet tall casts a shadow of 172 feet, what is the measure of the angle of elevation of the sun?

53.13° Signup and view all the answers

What is the effective annual yield of an investment with a nominal annual interest rate of 14.23% compounded monthly?

14.61% Signup and view all the answers

What is the value of x in the equation log a 2 = x and log a 3 = y?

1/2 Signup and view all the answers

What is the value of the common ratio in the infinite geometric series 2 + ¹/₂ + ¹/₈ + ...?

There isn't a common ratio Signup and view all the answers

Which of the following expressions is equivalent to (n + 2)! / (n)!?

(n + 2)(n + 1) Signup and view all the answers

What is the two-hundredth term of the sequence 1, 5, 9, 13,....?

797 Signup and view all the answers

What is the magnitude of the vector PQ where P = (12, 4) and Q = (-4, 6)?

2√65 Signup and view all the answers

If u = (8, -5) and v = (-10, -3), what is the value of v - u?

(-18, 2) Signup and view all the answers

What type of series is 32 + 16 + 8 + 4 + 2 + 1?

Finite geometric series Signup and view all the answers

What is the sum of the infinite series ∑(4/5)^i from i = 1 to infinity?

4 Signup and view all the answers

What is the value of u ⋅ v where u = (3, -5) and v = (4, 8)?

52 Signup and view all the answers

What is the sigma notation for the sum 12 + 15 + 18 + 21 + 24?

∑(12 + n) from i = 0 Signup and view all the answers

What is the unit vector in the direction of u = (5, -5)?

1/2, -1/2 Signup and view all the answers

What is the nth term of the sequence 15, 45, 135, 405, ...?

15(3)^(n-1) Signup and view all the answers

What is the angle between the vectors (4, 3) and (3, 5) to the nearest tenth of a degree?

157.8° Signup and view all the answers

What is the value of ∑(-1)²ᵏ from k = 1 to infinity?

1/3 Signup and view all the answers

What is the partial sum of the sequence 8, 32, 128, 512, ... up to the nth term?

5(4)^(n-1) - 8 Signup and view all the answers

What is the value of x in the equation 2^x = 16?

4 Signup and view all the answers

What is the formula for the sum of the first n terms of an arithmetic series?

n(a + l)/2 Signup and view all the answers

Study Notes

Vectors and Geometry

Given P(12,4) and Q(-4,6), the magnitude of the vector PQ is 2√65.
Let u = 8,-5 and v = -10,-3, then v - u = -18,2.
Let u = 3,-5 and v = 4,8, then u · v = -28.
Let u = 5,-5, the unit vector in the direction of u is 1/√2, -1/√2.
The angle between the vectors 4,3 and 3,5 is 22.2°.

Series and Sequences

The common ratio of the infinite geometric series 2 + 1/8 + 1/32 + ... is 1/4.
The expression equivalent to (n+2)! / (n)! is (n+2)(n+1).
The sum of the infinite series Σ(4/5)^i from i = 1 to infinity is 4/5.
The two-hundredth term of the sequence 1, 5, 9, 13, ... is 797.
The series 32 + 16 + 8 + 4 + 2 + 1 is a finite geometric series.
The sum of the infinite series Σ(4/5)^i from i = 1 to infinity is 4/5.
The sum 12 + 15 + 18 + 21 + 24 can be written in sigma notation as Σ(3n+9) from i = 0 to 4.
The nth term of the sequence 15, 45, 135, 405, ... is a_n = 15(3^(n-1)).

Exponential Functions

If log_a(2) = x and log_a(3) = y, then log_a(3/2) = y - x.
The domain of the function f(x) = -log2(x+1) is (-1, ∞).
If the graph of f(x) = 3^x is shifted 4 units to the right, the equation of the transformed graph is f(x) = 3^(x-4).

Applications

If Higgins wants to triple her money in 7.5 years, she needs to obtain an interest rate of 14.61% compounded monthly.
The number of United States citizens who traveled to foreign countries in 2005 can be modeled using an exponential model, and is approximately 66.2 million.

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Solve vector operations and find magnitude of vectors in different scenarios.