Slope Tutorial

Below is the series on Slope Tutorial. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

SLOPE01 Definition of Slope and the Slope Formula
SLOPE02 Finding the Slope of a Line
SLOPE03 Positive and Negative Slopes of a Line
SLOPE04 Zero and Undefined Slopes
SLOPE07 How to Find the Slope of a Line from the Equation Part 1
SLOPE08 How to Find the Slope of a Line from the Equation Part 2


You might also want to learn about Calculus click here:

Calculus Series – Part I
Calculus Series – Part II
Calculus Series – Part III

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Calculus Series – Part III

Below is the series on Calculus – Part III. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 93: Calculus 3.00 Introduction to Chapter 3
Lesson 94: Calculus 3.01 Formulas for Finding the Derivative of Trigonometric Functions
Lesson 95: Calculus 3.02 Derivative of Trigonometric Functions – Worked Examples 1
Lesson 96: Calculus 3.03 Derivative of Trigonometric Functions – Worked Examples 2
Lesson 97: Calculus 3.04 Derivative of Trigonometric Functions – Worked Examples 3
Lesson 98: Calculus 3.07 Introduction to Chain Rule
Lesson 99: Calculus 3.08 Chain Rule – Worked Examples 1
Lesson 100: Calculus 3.09 Chain Rule – Worked Examples 2
Lesson 101: Calculus 3.10 Chain Rule – Worked Examples 4 (Negative Exponents and Radicals)
Lesson 102: Calculus 3.11 Chain Rule – ‘Derivation’ of The General Power Rule

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Calculus Series – Part II

Below is the series on Calculus – Part II. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 51: Calculus 1.50 Definition of Continuity of a Function at a Number
Lesson 52: Calculus 1.51 Understanding the Definition of Continuity
Lesson 53: Calculus 1.52 Continuity at a Point Sample Problem 1 (Removable Discontinuity)
Lesson 54: Calculus 1.53 Sample Problem on Continuity Part 2 (Infinite Discontinuity)
Lesson 55: Calculus 1.54 Sample Problem on Continuity Part 3
Lesson 56: Calculus 1.55 Sample Problem on Continuity Part 4
Lesson 57: Calculus 1.56 Continuous and Discontinuous Functions – Summary
Lesson 58: Calculus 2.00 The Concept and Definition of Derivative
Lesson 59: Calculus 2.01 Finding the Slope of a Line Tangent to a Curve at Any Point
Lesson 60: Calculus 2.02 Finding the Slope of a Line Tangent to a Curve at a Particular Point

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Calculus Series – Part I

Below is the series on Calculus – Part I. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 1: Calculus 1.00 Overview of the Calculus Video Series
Lesson 2: Calculus 1.01 The Origins of Calculus
Lesson 3: Calculus 1.02 The Tangent Problem
Lesson 4: Calculus 1.03 The Area Problem
Lesson 5: Calculus 1.04 Finding the Slope of a Tangent Line Part 1
Lesson 6: Calculus 1.05 Finding the Slope of a Tangent Line Part 2
Lesson 7: Calculus 1.06 Finding the Slope of a Tangent Line Part 3
Lesson 8: Calculus 1.07 Finding the Slope of a Tangent Line Part 4
Lesson 9:: Calculus 1.08 Summary of the Tangent Problem
Lesson 10: Calculus 1.09 Another Notation for Slope of a Tangent Line

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Precalculus Tutorials for Senior High School and College Students

Below are the tutorials on Precalculus Tutorials for Senior High School and College Students. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 1.00: Introduction to Precalculus for Senior High School
Lesson 1.01: A Brief Review of the Cartesian Plane
Lesson 1.02: The Conic Sections – Circle, Ellipse, Parabola, Hyperbola
Lesson 1.03: Equation of a Circle with Center at the Origin – Practice Exercises
Lesson 1.04: Equation of a Circle with Center (h,k)
Lesson 1.05: Equation of a Circle with Center at the Origin and Radius
Lesson 1.06: Writing the Equation of a Circle with Center (h,k) and the Radius
Lesson 1.07: Finding the Center and Radius of a Circle Given its Equation
Lesson 1.08: Standard and General Forms of the Equation of a Circle
Lesson 1.09: Converting Equations of a Circle From Standard Form to General
Form

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May slope ba ang graph ng quadratic function?

Oo. Ang slope ng graph ng quadratic functions o kahit ano mang curve ay hindi pare-pareho. Hindi ito katulad ng slope ng mga straight lines o graph ng linear functions na constant.

Ang slope ng straight lines ay constant. Kung mayroong straight line, pwede tayo kumuha ng 2 points, at gumawa ng right triangle kagaya sa unang figure sa ibaba. Ang vertical segment ay ang rise, at ang horizontal segment ay ang run. Sa ibaba, ang rise ay 2 at run ay 1, kaya ang slope ng graph ay  \frac{2}{1} = 2.

Sa quadratic function naman, iba-iba ang slope sa bawat point sa curve. Para makuha natin ang slope sa isang particular point, kailangan nating mag-drawing ng line na tangent sa graph ng quadratic function at dumadaan sa napiling point.  Halimbawa, ang slope ng graph ng quadratic function sa point A sa ibaba ay equal sa slope ng tangent line na dumadaan dito.  Sa point B, ang slope nito ang slope ng line na dumadaan sa point B.    Tama kayo sa inyong iniisip. Ang bawat point sa curved graph ay may sariling slope.  Continue reading

Totoo ba na 0.999… =1?

Oo, 0.999 \cdots = 1.

Bago ko ipaliwanag ang sagot, nais kong bigyang diin na ang number na 0.999 \cdots ay nangangahulugang walang katapusan ang 9  (infinitely many 9’s) papuntang kanan ng decimal point. Ang proof nito ay napakadali lang.

Proof

Alam natin na ang \frac{1}{3} = 0.333 \cdots. I-multiply natin sa 3 ang both sides ng equation, at ang resulta ay 1 = 0.999 \cdots. \blacksquare

Marami pang proofs ang equation sa itaas, pero isang proof lang ay sapat na para masabi natin na ang isang equation (o theorem) ay totoo. Malamang ay nagrerebelde ang isip ninyo sa pagtanggap nito, ngunit kailangan ninyo magaral ng Elementary Calculus para ninyo ito lubos maintindihan. Para sa isa pang paliwaanag, ang 0.999 \cdots ay pwede natin gawing fraction na may infinitely many terms. Ito ay equal sa

\displaystyle\frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{9}{10000} + \cdots.

kung saan ang tatlong tuldok ay nangangahulugang walang katapusan ang number of terms sa kanan. Sa fractions at decimals, parehong makikita natin na habang padami ng padami ang 9’s, palapit ito ng palapit sa 1. Continue reading