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Calculus

rootmath via YouTube

Overview

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This course on Calculus aims to help learners understand and apply fundamental concepts such as limits, derivatives, and integrals. By the end of the course, students will be able to solve limit problems, find derivatives using various rules like the power rule and chain rule, identify critical points, determine concavity, optimize functions, calculate antiderivatives, and perform integration using techniques like u-substitution and natural logarithms. The course utilizes a combination of theoretical explanations, step-by-step examples, and problem-solving exercises to enhance understanding. This course is designed for individuals interested in building a strong foundation in calculus, particularly suitable for students studying mathematics, engineering, physics, or related fields.

Syllabus

Welcome to Calculus 1.
1.1 Introduction to Limits.
1.2 Estimating Limits.
1.3 Limits that Fail to Exist [01] - |x|/x.
1.3 Limit that Fails to Exist [02] - 1/x^2.
1.3 Limits that Fail to Exist [03] - sin(1/x).
1.4 Properties of LImits.
1.5 Solving Limits [01] (Factoring).
1.5 Solving Limits [02] (Rationalization).
1.5 Solving Limits [03] (Fractions).
1.6 Trig Limits [01] (1-cosx/x).
1.6 Trig Limits [02] (tanx/x & sin3x/x).
1.7 Limit Definition - Epsilon Delta [01] (NEW).
1.7 Limit Definition - Epsilon Delta [02].
1.7 Epsilon Delta Limit Definition [03] (example 1).
1.7 Proving a Limit: x^2 = 4 (advanced).
1.8 One Sided Limits.
1.8 One Sided Limit (example 1).
1.8 Continuity.
1.8 Continuity (example 1).
1.6 Trig Limits [03] Proof of sinx/x.
1.9 Geometric Interpretation of sec(x) and tan(x).
1.9 Geometry of [1 - cos(x)/x].
1.9 Problem Solving [01].
1.9 Problem Solving [02].
2.1 - Definition of the Derivative.
2.1 Finding the Slope of a Tangent Line - Example 1.
2.1 Finding the Slope of Tangent Line - Example 2.
2.1 Finding the Slope of a Tangent Line - Example 3.
2.2 Function vs. Derivative - Example 1.
2.2 Function vs. Derivative - Example 2.
2.2 Function vs. Derivative - Example 3.
2.3 Derivative of a Constant.
2.3 Power Rule.
2.4 Derivative of sin(x).
2.4 Derivatives - Trig Functions.
2.5 Product Rule.
2.6 Chain Rule (function notation).
2.6 Chain Rule (Leibniz notation).
2.6 Chain Rule - Example 1 - e^(2x).
2.6 Chain Rule - Example 2 - sin(x^2 + 1).
2.6 Chain Rule - Example 3 - Advanced.
2.7 Quotient Rule 01.
2.7 Quotient Rule 02.
2.8 Introduction to Implicit Differentiation.
2.8 Implicit Differentation (example 1).
2.8 Implicit Differentiation (example 2) - ln(x).
2.8 Derivative of arcsin(x).
2.8 Derivative of arcsec(x).
2.9 Related Rates Introduction.
2.9 Relates Rates Example 01 (Filling a Pool).
2.9 Related Rates Example 02 (Filling a Trough).
2.9 Related Rates Example 03 (Security Laser Part 1).
2.9 Related Rates Example 03 (Security Laser Part 2).
2.9 Related Rates Example 04 (Man walking with his shadow).
3.1 Introduction to Extrema.
3.1 Extrema Example.
3.1 Critical Numbers.
3.2 Finding Critical Numbers [Example 1].
3.2 Finding Critical Numbers [Example 2].
3.2 Finding Critical Numbers [Example 3].
3.2 Finding Critical Numbers [Example 4].
3.2 Finding Critical Numbers [Example 5].
3.3 Rolle's Theorem.
3.3 Mean Value Theorem.
3.3 Mean Value Thereom Example (prove a car was speeding).
3.4 First Derivative Test [Example 1].
3.4 First Derivative Test [Example 2] (Part 1).
3.4 First Derivative Test [Example 2] (Part 2).
3.5 Introduction to Concavity.
3.5 Concavity and the Second Derivative [1].
3.5 Inflection Points.
3.5 Concavity and the Second Derivative [2].
3.6 Optimization - Box with max volume (Part 1).
3.6 Optimization - Box with max volume (Part 2).
3.6 Optimization 02 (circle and square with maximum area).
3.7 Linear Approximation.
4.1 Introduction to Antiderivatives.
4.1 Antiderivative Power Rule.
4.1 Basic Properties of Antiderivatives.
4.1 Common Antiderivatives.
4.2 Intro to Area Under a Curve.
Summation Formulas and Sigma Notation (Part 1) Notation.
Summation Formulas and Sigma Notation (Part 2) Formulas.
Summation Formulas and Sigma Notation (Part 3) Advanced Properties.
Summation Formulas and Sigma Notation (Part 4) Examples.
4.2 Estimating the Area Under a Curve.
4.3 Exact Area Under A Curve.
4.3 Exact Area Under a Curve 02.
4.3 Exact Area - Left Hand Sum.
4.3 Exact Area Under a Curve 3.
4.4 Riemann Sum and the Definite Integral.
4.5 First Fundamental Thereom of Calculus.
4.5 First Fundamental Theorem of Calculus (Examples).
4.6 Properties of Integrals.
4.6 Area Under the x-axis.
4.6 Average Value of a Function.
5.1 Integration: Re-Writing an Integral - Ex.1.
5.1 Integration: Re-Writing an Integral - Ex.2.
5.1 Integration: Re-Writing an Integral - Ex.3.
5.2 Integration: U-Substitution - Ex.1.
5.2 Integration: U-Substitution - Ex.2.
5.2 Integration: U-Substitution - Ex.3.
5.2 Integration | U-Substitution - Ex. 4.
5.2 Integration | U-Substitution - Ex. 5.
5.2 Integration | U-Substitution - Ex.6.
5.3 Integration | Natural Log (ln) - Ex.1.
5.3 Integration | Natural Log (ln) - Ex.2.
5.3 Integration | Natural Log (ln) - Ex.3.
5.3 Integration | Natural Log (ln) - Ex. 4.
5.4 Integration | Integral of secx.
5.4 Integration | Integral of tanx.

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rootmath

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