Ch. 9 - Miscellaneous

Question 1

Definition of a Logarithm

log_bn = x ⇔ b^x = n

Answer 1

Change of Base Formula

log_BA = logA / logB

Question 2

Logs:

– The Product Rule

log_<em>b</em>(xy) = log_<em>b</em>x + log_<em>b</em>y

– The Quotient Rule

log_b(x/y) = log_bx – log_<em>b</em>y

Answer 2

– The Power Rule

log_<em>b</em>(x^r) = r•log_<em>b</em>x

Definitions of a Natural Logarithm

ln(n) = x ⇔ log_<em>e</em>n = x ⇔ e^x = n

Question 3

Answer 3

Question 4

Arithmetic Sequence

“one in which the differnece between successive terms is constant.”

Answer 4

To get to the nth term in a sequence, take (n – 1) steps from the first term.

Question 5

The nth Term of an Arithmetic Sequence

a_n = ?

Sum of the First n Terms of an Arithmetic Sequence

sum = ?

Answer 5

a_n = a₁ + (n – 1)d

sum = ***n *(a₁ + a_n / 2**)

a₁ = first term (starting value).

d = difference between any two successive terms.

Question 6

Geometric Sequence

formed by taking a starting value and multiplying it by the same factor again and again.

Answer 6

Again, it takes (n – 1) steps to get the nth term.

Question 7

The nth Term of a Geometric Sequence

a_n = ?

Sum of the First n Terms of a Geometric Sequence

sum = ?

Answer 7

a_n = a₁r^{n – 1}

sum = a₁(1 – rⁿ) / (1 – r)

Question 8

The sum of an infinite geometric series can be calculated only when the constant factor is between -1 and 1.

Answer 8

Sum of an Infinite Geometric Sequence

sum = a₁ / (1 – r) for -1 < r < 1

Question 9

Adding vectors

Answer 9

Subtracting vectors

Question 10

You can add or subtract two vectors by adding or subtracting their x and y components.

Answer 10

For example, if vector u has components (1, 3) and vector v has components (-1, 5), then the resulting vector u + v would have components

(1 + (-1), 3 + 5) = (0, 8).

Question 11

Logic – The Contrapositive

Given the statement A → B, you also know ¬B → ¬A.

Answer 11

If…

stone is precious → stone harder than glass

then the only thing for certain…

stone not harder than glass → stone not precious

Question 12

Imaginary numbers

Answer 12

• i* = √(-1)
• i*² = -1

Question 13

Powers of i

Answer 13

Question 14

The complex plane

A complex number is a specific find of imgainary number—specifically, the sum of a real number and an imaginary number. A complex number is one that takes the form a + b**i, where a and b are real numbers and i is the imaginary unit, the sqaure root of -1.

Answer 14

Just plot a, the real component of the complex number, on the x-axis; and bi, the imaginary component, on the y-axis.

Question 15

The distance between a complex number and the origin is most often refered to as the magnitude and orabsolute value of a complex number.

Answer 15

Question 16

Polymonial divison

Answer 16

PLUG IN!!!

When you Plug In on ploynomial divison questions that ask for a remainder, you’ll find that bigger numbers, such as 10, are better.

Question 17

The determinant of the 2 x 2 matrix

Answer 17

Two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.

Question 18

Adding/Subtracting Matrices

Answer 18

Multiplying Matrices