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Math

It can be valuable to a viewer to have a video divided into parts/sections/segments (chapters) so that they can jump around to important moments in the video. The sectioning for this video follows:

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#howtoaddchapters #youtubechapters #addchapters

When working with a normal distribution you are provided a mean and standard deviation. The empirical rule tells us that there is 68%, 95%, and 99.7% of the normal distribution contained in the first, second, and third standard deviation away from the mean, respectively.

Where do these numbers come from?

In this video, an overview about integrals and the relationship to the standard distribution probability density function provide insight into where these numbers come from.

Some images made using Geogebra.

0:00 What does the empirical rule tell us?

1:05 Normal Distribution probability density function.

1:45 Normal curve with mean of 0 and standard deviation of 1.

2:20 Probability at point vs probability of interval.

2:52 Using integral to calculate area under normal curve.

3:11 Integrals to compute area between standard deviation of the mean for N(0,1) example.

4:29 Setting up table to compare areas for different normal curves.

5:01 How does changing the standard deviation change the spread of the curve?

5:40 Integrals to compute area between standard deviation of the mean for N(0,0.5) example.

6:35 Integrals to compute area between standard deviation of the mean for N(50,10) example.

7:30 Integrals to compute area between standard deviation of the mean for N(1000,25) example.

7:45 Conversation about how area under curve didn't change when varying the mean and standard deviation.

8:12 The empirical rule saves you time from recalculating integrals over and over again.

#68-99-99.7

#empiricalrule

#normaldistribution

#JoeCMath

The value of b^2-4ac gives us some immediate information about how the graph of our Quadratic Function will look. This video discusses the three scenarios one will run into when trying to graph a Quadratic Function.

Some images made using Geogebra.

#b^2-4ac

#Quadratics

#Discriminant

#JoeCMath

#Algebra

0:00 Introduction

0:23 Three scenarios for b^2-4ac

0:40 b^2-4ac is positive

1:44 b^2-4ac is zero

2:40 b^2-4ac is negative

3:32 Summary

4:35 Subscribe if the Krusty Krab is better than the Chum Bucket

Converting a standard form quadratic function to a graph can be accomplished by finding the y-intercept, vertex, and x-intercepts (if they exist) and fitting a parabola to those points. It is easy to find these points when a quadratic function is in Standard Form because they can all be found by plugging a, b, and c into particular formulas.

y-intercept: (0,f(0))

vertex: (-b/2a,f(-b/2a))

x-intercepts: use quadratic formula

Some images made using Geogebra.

#StandardForm

#GraphingQuadratic

#GraphParabola

#StandardFormtoGraph

#Algebra

#JoeCMath

0:00 Introduction

0:35 What the value of "a" tells you

0:40 General points we can get from stand form quadratic

1:30 Example 1

5:08 Example 2

7:16 Pr(Joe Happy Today) LEQ Pr(You Subscribed) LEQ 1

Polynomial division (long division/ algebraic division) is similar to the long division process for two numbers. A small review of the long division format occurs at the beginning followed by two examples (one where the remainder is zero and another where the remainder is non-zero).

These examples were selected to motivate the connection between factoring polynomials of degree greater than two to the polynomial division process.

0:00 Introductory review

0:53 Example 1 starts

5:59 Example 2 starts

9:22 Small takeaway

#Polynomial #LongDivision #PolynomialDivision

#JoeCMath

U substitution (Integration by Substitution) is a common approach to solving integrals that contain a composition of functions. A function is contained within another function with its derivative nearby. These type of problems could be solved using U-Substitution.

0:00 Introduction

0:24 Derivative Chain Rule Revisited

1:00 Example : Find the integral of (x^5+1)^3*5x^4dx using u-substitution.

2:54 U-Substitution Steps

3:14 Example : Find the integral of (3x-1)^6dx using u-substitution.

5:02 Example : Find the integral of ((sin(ln(x)))/x)dx using u-substitution.

7:20 Thanks for watching!

#usubstitution

#Calculus

#JoeCMath

Finding the derivative of a function that contains the division of two different function is easy if you know how to use the quotient rule. This video provides the structure for the quotient rule and presents to examples of it's use.

#QuotientRule

#Derivatives

#QuotientRuleExamples

#JoeCMath

Introduction: (0:00)

General quotient rule: (0:14)

Example 1: (0:32)

Example 2: (2:10)

Closing comments: (3:14)

SUBSCRIBE PLEASE (4:12)

A short introduction to the chain rule as well as three examples.

Two examples include functions that only contain two nested functions. The final example goes over how to use the chain rule when a function contains three or more nested functions.

#ChainRule

#Derivatives

#ChainRuleExamples

#DerivativeExamples

#JoeCMath

0:00 Introduction

0:34 Example 1

1:28 Example 2

2:27 How to handle a function with multiple layers

4:42 Joe likes when you subscribe!

The process of completing the square for a quadratic expression, equation or, function can be hard to complete. Depending on the instructions you may have to also graph the function once it is in vertex form, or find the x-intercepts using vertex form.

This video goes through three examples with various levels of difficulty.

#CompletingTheSquare

#CompleteTheSquare

#StandardtoVertexForm

#Algebra

#JoeCMathExamples

0:00 Introduction

0:15 Refresher on process. General beside example.

1:14 Easy example.

2:34 Medium example.

4:25 Hard example.

7:21 A subscription to Joe implies absolutely nothing.

Anytime you have an expression that contains multiple variables raised to powers and you are told to simplify the expression completely, you need to be aware of all the rules for exponents.

When dividing variables that contain a common base we use the following rule to simplify:

x^m/x^n=x^(m-n)

#Exponents

#Division

#ExponentRules

#JoeCMath

#Algebra

0:00 Introduction

0:27 Where the rule comes from

1:40 Example 1

2:00 Example 2

2:30 Talk about relationship between m and n

3:12 Example 3

4:35 Fractions with + or - warning

5:50 Subscribe for more videos!

Indefinite and definite integral problems begin the same way but differ in how you interpret their results.

The result of an indefinite integral is a family of functions whereas a definite integral determines the area underneath the function you are integrating over the closed interval [a,b].

0:00 Introduction

0:21 Indefinite Integral

1:15 Definite Integral

2:53 Checking out examples on wolfram alpha

4:15 Review of definite vs indefinite integral results

#IndefiniteIntegral

#DefiniteIntegral

#AreaUnderCurve

#Calculus

#JoeCMath

Introduction to the power rule for derivatives. The power rule is

d/dx[x^n]=n*x^(n-1)

#PowerRule

#Derivatives

#PowerRuleExamples

#DerivativeExamples

#JoeCMath

0:00 Introduction

0:40 Example 1

1:29 Example 2

2:21 Detecting multiple rules in derivative problems

3:26 Subscribe to bring Joe more POWER

Tables found at: https://en.wikibooks.org/wiki/....Engineering_Tables/N

Z-scores are used to relate any normal distribution to the standard normal distribution (mean =0, standard deviation =1). Once you have a z-score, you have to use a z-score table in order to calculate probabilities or areas underneath the standard normal curve.

This video focuses on how to use a z-score table to answer simple questions about z-scores and the percent of the distribution contained below, above, or between given z-scores.

Normal distribution images made using Geogebra.

0:00 Introduction

0:08 Z-score and z-score table

0:55 Area below a z-score example

1:55 Area above a z-score example

3:25 Area between two z-scores example

#zscore

#zscoretable

#normaldistribution

#standardnormal

#joecmath

#statistics

Factor by grouping can be rather hard. Hopefully, after watching this video, you will feel a little more familiar with the process. It can be easy to mix up all of the different ways one can factor a quadratic expression, equation, or function.

Factor by grouping examples can be found in: "https://youtu.be/R-44aw7Wzv0"

#FactorByGrouping

#Process

#Algebra

0:00 Introduction

0:43 How to identify a, b, and c values

2:07 Splitting up the bx term

2:25 Factor by grouping example

6:21 Review of factor by grouping process

6:54 Takeaways

7:18 I'm Joe, and I'd like you to subscribe.

Matrix addition and subtraction is possible when the two matrices you are attempting to add or subtract have the same number of rows and columns.

0:00 Introduction

0:12 Talk about rows and columns agreeing

0:47 Matrix Addition Example

2:10 Matrix Subtraction Example

2:57 Review of topic

#addmatrix

#subtractmatrix

#matrices

#joecmath

The 68 - 95 -99.7 Rule (Empirical Rule) lets us know approximately what percent of the normal distribution exists between the first, second, and third stand deviation of the mean, respectively.

Combining this information with the fact that the normal distribution is symmetric about the mean, we can calculate the percent of the distribution that falls between each standard deviation. This video reminds you how to come up with these percentages on the fly if all you can remember is the 68 - 95 -99.7 Rule (Empirical Rule).

Some images made using Geogebra.

0:00 Introduction

0:25 Percent between first standard deviation of the mean

0:57 Percent between second standard deviation of the mean

1:35 Percent between third standard deviation of the mean

2:01 Percent in tails to infinities

#standarddeviation

#normaldistribution

#joecmath

#statistics

Videos supplement material from the textbook Physics for Engineers and Scientist by Ohanian and Markery (3rd. Edition) (http://books.wwnorton.com/book....s/Physics-for-Engine Feel free to post questions and/or suggestions in the comments and I'll respond if possible. The videos are mostly for my students, but since others seem to find them useful I'll help as much as I possibly can!

The "a" value of a quadratic function in standard form tells us whether our parabola opens upwards (concave up) or downwards (concave down).

If a is positive (a is greater than 0) then the parabola will open upwards.

If a is negative (a is less than 0) then the parabola will open downwards.

#ConcaveUp/Down

#Quadratics

#Algebra

0:00 Introduction

0:11 What positive a tells us

0:25 What negative a tells us

0:48 Positive a examples

0:59 Negative a examples

1:20 Hey you, you should subscribe!

SOHCAHTOA is a useful way to remember the relationship between the sides of a right triangle and the trig functions sine, cosine, and tangent of a particular angle within the triangle.

0:00 Intro

0:25 SOHCAHTOA Overview

2:07 Example

3:00 Thanks for watching!

#SOHCAHTOA

#Trig

#JoeCMath