Embark on a comprehensive journey through Single Variable Calculus in this 7-hour playlist covering limits, derivatives, and the fundamentals of integrals. Begin with the velocity problem, exploring it numerically and graphically, before delving into the concept of limits. Master limit laws, algebraic tricks, and continuous functions. Explore the Intermediate Value Theorem and piecewise functions. Progress to derivatives, learning their definition, rules, and applications, including the power rule, product and quotient rules, and chain rule. Tackle implicit differentiation, logarithmic differentiation, and related rates. Discover linear approximations, the Mean Value Theorem, and optimization problems. Grasp L'Hopital's Rule and its applications. Transition to anti-derivatives and definite integrals, understanding Riemann sums and the Fundamental Theorem of Calculus. Practice integration techniques, including substitution methods and back substitution. Conclude with the average value of continuous functions and a comprehensive exam walkthrough, solidifying your understanding of this essential mathematical foundation.
Overview
Syllabus
The Velocity Problem | Part I: Numerically.
The Velocity Problem | Part II: Graphically.
A Tale of Three Functions | Intro to Limits Part I.
A Tale of Three Functions | Intro to Limits Part II.
What is an infinite limit?.
Limit Laws | Breaking Up Complicated Limits Into Simpler Ones.
Building up to computing limits of rational functions.
Limits of Oscillating Functions and the Squeeze Theorem.
Top 4 Algebraic Tricks for Computing Limits.
A Limit Example Combining Multiple Algebraic Tricks.
Limits are simple for continuous functions.
Were you ever exactly 3 feet tall? The Intermediate Value Theorem.
Example: When is a Piecewise Function Continuous?.
Limits "at" infinity.
Computing Limits at Infinity for Rational Functions.
Infinite Limit vs Limits at Infinity of a Composite Function.
How to watch math videos.
Definition of the Derivative | Part I.
Applying the Definition of the Derivative to 1/x.
Definition of Derivative Example: f(x) = x + 1/(x+1).
The derivative of a constant and of x^2 from the definition.
Derivative Rules: Power Rule, Additivity, and Scalar Multiplication.
How to Find the Equation of a Tangent Line.
The derivative of e^x..
The product and quotient rules.
The derivative of Trigonometric Functions.
Chain Rule: the Derivative of a Composition.
Interpreting the Chain Rule Graphically.
The Chain Rule using Leibniz notation.
Implicit Differentiation | Differentiation when you only have an equation, not an explicit function.
Derivative of Inverse Trig Functions via Implicit Differentiation.
The Derivative of ln(x) via Implicit Differentiation.
Logarithmic Differentiation | Example: x^sinx.
Intro to Related Rates.
Linear Approximations | Using Tangent Lines to Approximate Functions.
The MEAN Value Theorem is Actually Very Nice.
Relative and Absolute Maximums and Minimums | Part I.
Relative and Absolute Maximums and Minimums | Part II.
Using L'Hopital's Rule to show that exponentials dominate polynomials.
Applying L'Hopital's Rule to Exponential Indeterminate Forms.
Ex: Optimizing the Volume of a Box With Fixed Surface Area.
Folding a wire into the largest rectangle | Optimization example.
Optimization Example: Minimizing Surface Area Given a Fixed Volume.
Tips for Success in Flipped Classrooms + OMG BABY!!!.
What's an anti-derivative?.
Solving for the constant in the general anti-derivative.
The Definite Integral Part I: Approximating Areas with rectangles.
The Definite Integral Part II: Using Summation Notation to Define the Definite Integral.
The Definite Integral Part III: Evaluating From The Definition.
"Reverse" Riemann Sums | Finding the Definite Integral Given a Sum.
Fundamental Theorem of Calculus 1 | Geometric Idea + Chain Rule Example.
Fundamental Theorem of Calculus II.
Intro to Substitution - Undoing the Chain Rule.
Adjusting the Constant in Integration by Substitution.
Substitution Method for Definite Integrals **careful!**.
Back Substitution - When a u-sub doesn't match cleanly!.
Average Value of a Continuous Function on an Interval.
Exam Walkthrough | Calc 1, Test 3 | Integration, FTC I/II, Optimization, u-subs, Graphing.
♥♥♥ Thank you Calc Students♥♥♥ Some final thoughts..
Taught by
Dr. Trefor Bazett
