This course on Calculus 1 aims to teach students the fundamental concepts and techniques of differential calculus. By the end of the course, learners will be able to understand and apply limits, continuity, derivatives, and integrals. They will also learn various rules for differentiation, including the power rule, product rule, quotient rule, and chain rule. The course covers applications of derivatives such as related rates, linear approximation, and optimization. Additionally, students will be introduced to antiderivatives, the fundamental theorem of calculus, and techniques for finding areas under curves. The teaching method includes theoretical explanations, proofs of key theorems, and numerous examples and exercises to reinforce learning. This course is intended for individuals interested in building a strong foundation in calculus, particularly suitable for students pursuing studies in mathematics, engineering, physics, or related fields.
Overview
Syllabus
Graphs and Limits.
When Limits Fail to Exist.
Limit Laws.
The Squeeze Theorem.
Limits using Algebraic Tricks.
When the Limit of the Denominator is 0.
Limits at Infinity and Graphs.
Limits at Infinity and Algebraic Tricks.
Continuity at a Point.
Continuity Example with a Piecewise Defined Function.
Continuity on Intervals.
Continuity and Domains.
Intermediate Value Theorem.
Derivatives and Tangent Lines.
Computing Derivatives from the Definition.
Interpreting Derivatives.
Derivatives as Functions and Graphs of Derivatives.
Proof that Differentiable Functions are Continuous.
Power Rule and Other Rules for Derivatives.
Higher Order Derivatives and Notation.
Derivative of e^x.
Proof of the Power Rule and Other Derivative Rules.
Product Rule and Quotient Rule.
Proof of Product Rule and Quotient Rule.
Special Trigonometric Limits.
Derivatives of Trig Functions.
Proof of Trigonometric Limits and Derivatives.
Derivatives and Rates of Change (Rectilinear Motion).
Marginal Cost.
The Chain Rule.
More Chain Rule Examples and Justification.
Justification of the Chain Rule.
Implicit Differentiation.
Derivatives of Exponential Functions.
Derivatives of Log Functions.
Logarithmic Differentiation.
Inverse Trig Functions.
Derivatives of Inverse Trigonometric Functions.
Related Rates - Distances.
Related Rates - Volume and Flow.
Related Rates - Angle and Rotation.
Maximums and Minimums.
Mean Value Theorem.
Proof of Mean Value Theorem.
Derivatives and the Shape of the Graph.
First Derivative Test and Second Derivative Test.
Derivatives and the shape of the graph - example.
Extreme Value Examples.
Linear Approximation.
The Differential.
L'Hospital's Rule.
L'Hospital's Rule on Other Indeterminate Forms.
Newtons Method.
Antiderivatives.
Finding Antiderivatives Using Initial Conditions.
Any Two Antiderivatives Differ by a Constant.
Summation Notation.
Approximating Area.
The Fundamental Theorem of Calculus, Part 1.
The Fundamental Theorem of Calculus, Part 2.
Proof of the Fundamental Theorem of Calculus.
The Substitution Method.
Why U-Substitution Works.
Average Value of a Function.
Proof of the Mean Value Theorem for Integrals.
Recitation 2 a solution and some hints.
Limit as x goes to infinity recitation problem.
Taught by
Linda Green
