Algebra and Geometry Review – Exercise 04

QUESTION 1: Factoring a perfect square trinomial with leading coefficient 1

                Factor: y^2 - 12x + 36

               

EXPLANATION: When factoring a polynomial, the following formulas are sometimes useful

1.       a^2 + 2ab + b^2 = (a + b)^2

2.        a^2 - 2ab + b^2 = (a - b)^2

We can use formula 2. to factor the given perfect square polynomial.

y^2 - 12x + 36 = y^2 – 2(y)(6) + 6^2

                = (y – 6)^2

ANSWER: (y – 6)^2

QUESTION 2: Simplifying a ratio of multivariate monomials

                Simplify: 25xy / 35yz

EXPLANATION: We simplify 25xy / 35yz as follows

                25xy / 35yz = 5xy / 7yz                   Canceling the common factor 5

                = 5x / 7z               Canceling the common factor y

ANSWER: 5x / 7z

Algebra and Geometry Review – Exercise 03

QUESTION 1: Evaluating an expression with a negative exponent

Rewrite the following without an exponent: 8^-2

EXPLANATION:

For any nonzero number a and any whole number n, we have the following.

Rule 1:                  a^-n = 1/a^n

Move a^-n to the denominator and change -n to n

Rule 2:                  1/a^-n = a^n

Move a^-n to the numerator and change -n to n

We need to rewrite 8^-2 without an exponent:

To do this, we first use Rule 1 and move 8^-2 to the denominator, making the exponent positive and evaluate

8^-2 = 1/8^2       using Rule 1

= 1/64                   Since 8^2 = 64

ANSWER: 1/64

Algebra and Geometry Review – Exercise 02

QUESTION 1: introduction to the quotient rule of exponents

Simplify y^7/y^4

EXPLANATION: The exponents tell us how many y's to multiply

                y^7/y^4 = Y*y*y*y*y*y*y / y*y*y*y

                = y*y*y / 1          Canceling gives us the following

                = y^3

ANSWER: y^3

QUESTION 2: Multiplying binomials with leading coefficients of 1

Multiply and Simplify your answer: (u-2)(u+7)

EXPLANATION:

                We want to remove the parentheses from the product (u-2)(u+7)

                We first multiply each term in the first factor (u-2) by each term in the second factor (u+7) using FOIL (First, Outer, Inner, Last).

                F:            Multiply the two First terms: u*u = u^2

                O:           Multiply the two Outside terms: u*7 = u7

                I :            Multiply the two Inside terms: -2*u = -2u

                L:             Multiply the two Last terms: -2*7 = -14

               

The product (u-2)(u+7) is then equal to the sum of these terms:

                (u-2)(u+7) = u^2 + 7u + - 2u – 14

                = u^2 + 5u -14

ANSWER: u^2 + 5u -14

Algebra and Geometry Review – Exercise 01

QUESTION 1: Using distribution and combining like terms to simplify: 3(y + 5) - 6y

EXPLANATION

                3(y + 5) - 6y = 3y + 15 – 6y            Using the distributive property to remove parentheses

                = 3y + 15 + (- 6y)               Writing subtraction as addition of a negative

                = 3y + (- 6y) +15                Using the commutative property to rearrange terms

                = - 3y + 15           Combining like terms

ANSWER: -3y + 15

QUESTION 2: Use the distributive property to remove the parentheses (-2 + 4x + 4v) (-7)

EXPLANATION: We use the distributive property as follows

                (-2 + 4x + 4v) (-7) = (-2)(-7) + (4x)(-7) + (4v)(-7)

                = 14 + (-28x) + (-28v)

                = 14 - 28x – 28v

ANSWER: 14 - 28x – 28v

Sensitivity Analysis | Statistics

This lesson we learn:
- Sensitivity and specificity: Quantifying whether tests for diseases are any good
- Predictive value: If you test positive/negative, what are your chances?
- Cutoffs and ROCs: Sensitivity vs. specifity, cutoff selection, and ROC curves
- Prevalence: How rare a disease is, and how it affects PPV and NPV

Positive predictive value (PPV), negative predictive value (NPV)
Receiver operating characteristic (ROC) curves
A rare disease has a prevalence near 0.
The more common a disease, the greater the prevalence.

Tangent lines - Drawing derivatives | Calculus

Tangent lines: Finding the "slope" of a curve
Drawing derivatives: Learn what a derivative is, and draw your own

Tangent lines - Drawing derivatives | Calculus:

Limits | Calculus

In this lesson we learn:
- Limits to infinity: What happens to functions as x gets really, really big?
- Vertical limits: Sometimes y goes off to infinity
- Finite limits: Limits when both x and y stay finite
- One-sided limits: Limits that come from only the left or right side
- Continuity (for real): Dive deeper into continuity, using one-sided limits
- Splitting limits: Tricks for simplifying limits (and when they don't work!)

In this section we will learn about limits whose value is infinity or minus infinity. 
The function f(x) will have a horizontal asymptote at y=L if either of the following are true; limit x go to infinity f(x) = L and limit x go to minus infinity f(x) = L.
The function f(x) will have a vertical asymptote at x = a if we have any of the following limits at x = a; limit x approaching a-negative f(x) = positive/negative infinity, limit x approaching a-positive f(x) = positive/negative infinity, and limit x approaching a f(x) = positive/negative infinity.
We say that the limit of f(x) is L as x approaches a and write this as limit x approaches a f(x) = L, provided we can make f(x) as close to L as we want for all x sufficiently close to a, from both sides, without actually letting x be a.