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Calculus • Series

# Comprehensive Calculus: Limits, Derivatives, and Integrals

Jose Roberto Cossich G

This series ended on July 24, 2023. All 1:1 and group chats related to this series are disabled 7 days after the last session.

### Series Details

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

✋ ATTENDANCE POLICY

I try to make the session times as accommodating as possible, however make-up sessions are always available. Please feel free to ask for one if needed!

Dates

March 26 - July 24

Learners

30 / 40

Total Sessions

39

Hey, my name's Jose, and I'm currently a senior. I'm about the biggest Indie music fan there is, I love 80s movies (John Hughes goat), and reading in my spare time. I enjoy math because it allows me to express myself artistically, and tutor for the mere sake of those aha moments. I've taken Multivariable Calc, Linear Algebra, Real and Complex Analysis, and I'm currently taking AP Physics C.

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