24 Questions
If Higgins invests $2500, how much will she have after 7.5 years if the interest rate is 14.23% compounded monthly?
7500
What is the value of log a 6 in terms of x and y, given log a 2 = x and log a 3 = y?
x + y
What is the domain of f(x) = -log2(x + 1)?
(-1, ∞)
What is the equation of the graph of f(x) = 3x shifted 4 units to the right?
f(x) = 3(x - 4)
Using an exponential model, what is the number of travelers (in millions) to foreign countries in 2005?
66.1
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°
What is the effective annual yield of an investment with a nominal annual interest rate of 14.23% compounded monthly?
14.61%
What is the value of x in the equation log a 2 = x and log a 3 = y?
1/2
What is the value of the common ratio in the infinite geometric series 2 + ¹/₂ + ¹/₈ + ...?
There isn't a common ratio
Which of the following expressions is equivalent to (n + 2)! / (n)!?
(n + 2)(n + 1)
What is the two-hundredth term of the sequence 1, 5, 9, 13,....?
797
What is the magnitude of the vector PQ where P = (12, 4) and Q = (-4, 6)?
2√65
If u = (8, -5) and v = (-10, -3), what is the value of v - u?
(-18, 2)
What type of series is 32 + 16 + 8 + 4 + 2 + 1?
Finite geometric series
What is the sum of the infinite series ∑(4/5)^i from i = 1 to infinity?
4
What is the value of u ⋅ v where u = (3, -5) and v = (4, 8)?
52
What is the sigma notation for the sum 12 + 15 + 18 + 21 + 24?
∑(12 + n) from i = 0
What is the unit vector in the direction of u = (5, -5)?
1/2, -1/2
What is the nth term of the sequence 15, 45, 135, 405, ...?
15(3)^(n-1)
What is the angle between the vectors (4, 3) and (3, 5) to the nearest tenth of a degree?
157.8°
What is the partial sum of the sequence 8, 32, 128, 512, ... up to the nth term?
5(4)^(n-1) - 8
What is the value of x in the equation 2^x = 16?
4
What is the formula for the sum of the first n terms of an arithmetic series?
n(a + l)/2
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.
Solve vector operations and find magnitude of vectors in different scenarios.
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