Finally! The long-awaited series in Calculus is here.
Main resources being used: AoPS and Khan Academy
Pre-requisites: Trigonometry, Algebra 1, Algebra 2, and Precalculus (recommended but not required).
The session descriptions are posted below, so please check them out!
This is the first of three series I will be hosting, covering Calculus 1, 2, and 3. After each series ends I will host the next.
In this course we will be covering the following:
LIMITS AND CONTINUITY (Unit 1)
- Introduction to limits
- One and two-sided limits
- Creating tables to approximate limits; approximating limits using tables
- Limits of composite functions
- Evaluating limits through direct substitution
-Limits of trigonometric and piecewise functions
-Evaluating limits by rationalizing and by using trigonometric identities
-Squeeze theorem and applications
-Introduction to continuity and types of discontinuities
-Continuity at a point and over an interval
-Removing discontinuities
-Limits at infinity and limits at infinity of quotients
-The Intermediate Value Theorem
-The formal epsilon-delta definition of a limit
DERIVATIVES (Unit 2):
-Introduction to derivatives and notation
-Average rate of change and the secant line
-Instantaneous rate of change and the tangent line
-The formal and informal definition of a derivative using limits
-Equation of the tangent line of a function at a point
-Estimating derivatives graphically and algebraically
-Differentiability at a point and over an interval (connection to continuity)
-Evaluating derivatives using the power rule, sum rule, product rule, quotient rule, and chain rule (including proofs of all the rules)
-Derivative of sin(x), cos(x), tan(x), csc(x), sec(x), cot(x), ln(x), e^x, a^x, and log_a(x) (including a derivation of all of the formulas)
-Derivative of inverse functions and inverse trigonometric functions
-Second derivatives, implicit differentiation, and related rates
-Position, velocity, and acceleration problems
-Local linearity and linear approximations to functions
-L'Hôpital's rule and the special case of L'Hôpital's rule
-The Mean Value Theorem for derivatives, the Extreme Value Theorem, and Rolle's Theorem
-Finding critical points, local minima and maxima, increasing and decreasing intervals of a function
-Finding absolute extrema over closed intervals and over the entire domain of a function
-Introduction to concavity
-The Second Derivative Test and points of inflection
-Optimization
INTRODUCTION TO INTEGRATION (Unit 3)
-introduction to accumulation of change
-Left and right Riemann sums; over and under-approximation
-Midpoint and trapezoidal sums
-Definition of indefinite integral using as the limit of Riemann sums
-The Fundamental Theorem of Calculus (including the proof), antiderivatives, and definite integrals
-Interpretation of accumulation functions: negative definite integrals, definite integrals over a single point; graphical interpretation and evaluation.
-Integrating sums of functions, switching the bounds of integration, and with functions as bounds.
-The reverse power rule, u-substitution, and integration by parts
-Antiderivatives of all the previously mentioned functions
-Improper integrals
