Explore the fundamentals of calculus in this comprehensive 7-hour course. Begin with an introduction to Zeno's Paradox and delve into the definition of limits, including right- and left-hand limits. Examine vertical asymptotes, limit laws, and the Squeeze Theorem. Study continuity, the Intermediate Value Theorem, and rates of change. Master the limit definition of derivatives and learn various derivative rules, including power, sum/difference, product, and quotient rules. Investigate derivatives of trigonometric, exponential, and logarithmic functions. Tackle implicit differentiation, logarithmic differentiation, and derivatives of inverse trigonometric functions. Explore function extrema, concavity, and graph sketching techniques. Apply L'Hospital's Rule to various indeterminate forms and solve optimization problems involving closest points, largest rectangles, boxes, and cone cups.
Overview
Syllabus
Intro to Calculus: Zeno's Paradox.
Definition of the Limit.
Right- and Left-hand Limits.
tan(x) & Vertical Asymptotes.
More Vertical Asymptotes: sec^2(x).
Asymptotes of a Rational Function.
A Nasty Rational Function.
The Limit Laws.
Limit Laws & Polynomials.
Limit Laws & Rational Functions.
Squeeze Theorem.
Continuity: 3 ways it fails.
Continuity: Two Examples.
Continuity: Properties.
Intermediate Value Theorem.
Rate of Change & Tangent Line.
Limit Definition of Derivative.
The Derivative as a Function.
Derivative of Square Root.
Derivative: Power Rule.
Derivative: Sum/Difference Rule.
Derivative of e^x.
Derivative: Product Rule.
Derivative: Quotient Rule.
Derivative Example Requiring Both Quotient and Product Rule.
Derivative of sin(x).
17th derivative of sin(x).
Derivative of tan(x) -- and the other trig functions.
Trig derivative example.
Motivating Chain Rule.
Justifying the Chain Rule.
Chain Rule Examples.
Derivative of a^x.
Intro to Implicit Differentiation.
Implicit Differentiation Example.
Second Derivative Using Implicit Differentiation.
Derivative of ln(x).
Logarithmic Differentiation.
Derivative of Arcsine.
Derivatives of Inverse Trig Functions.
Extrema of Function.
Extreme Value Theorem.
First Derivative Test.
Second Derivative and Concavity.
Sketching the Graph of a Polynomial.
Sketching Graph of Rational Function.
L'Hospital's Rule: Zero Over Zero?.
L'Hospital's Rule: Infinity Over Infinity?.
L'Hospital's Rule: Infinity Minus Infinity?.
L'Hospital's Rule: Zero To The Zero?.
Optimization: Closest Point.
Optimization: Largest Rectangle in Semicircle.
Optimization: Largest Box.
Optimization: Largest Cone Cup.
Taught by
math at andrews
