# Comprehensive Calculus: Limits, Derivatives, and Integrals

Next session on May 29, 2023

Hosted by Jose Roberto Cossich G

### Series Details

About

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 - June 16

Learners

33 / 40

Total Sessions

28

### About the Tutor

Hi, my name's Jose, I'm a Junior and an international student. I love listening to Indie music, watching 80s movies, 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'm currently taking Calc 3 and AP Physics C, hope to see you soon!

### Upcoming Sessions

Session 18

#### Differentiation: definition and basic derivative rules

(Session time: 1h30m)

To wrap up the second half of our proofs section of this second unit, we'll spend the first 20 minutes of today's session practicing some of the rules which we learnt yesterday, and then go on to derive the derivatives of sin(x), cos(x), tan(x), csc(x), sec(x), cot(x), ln(x), e^x, a^x, and log_a(x)! We'll probably spend another 30 minutes practicing applying these. Don't worry if that doesn't seem like enough--we'll spend the rest of the series practicing these!

Session 19

#### Contextual applications of differentiation

(Session time: 1h30m)

To wrap up the second half of our proofs section of this second unit, we'll spend the first 20 minutes of today's session practicing some of the rules which we learnt yesterday, and then go on to derive the derivatives of sin(x), cos(x), tan(x), csc(x), sec(x), cot(x), ln(x), e^x, a^x, and log_a(x)! We'll probably spend another 30 minutes practicing applying these. Don't worry if that doesn't seem like enough--we'll spend the rest of the series practicing these!

(Session time: 1h20m)

Now that we've gone over all the "rules" regarding derivatives, as well as some of the common applications of derivatives, we can focus on the principal role derivatives play: allowing us to find the minimum and maximum of functions. In order to do that, though, we must first come up with some really important results:

-L'Hôpital's rule and the special case of L'Hôpital's rule

-The Mean Value Theorem for derivatives -The Extreme Value Theorem -Rolle's Theorem We'll prove L'Hôpital's rule and its special case, and the proof of the other three will be provided in writing. (Session time: 1h) Trust me, the wait has been worth it. Today we'll be covering how to do the following: -Find critical points, local minima and maxima, increasing and decreasing intervals of a function -Find absolute extrema over closed intervals and over the entire domain of a function

Session 20

#### Contextual applications of differentiation

(Session time: 1h30m)

In single-variable calculus, there are two ways of finding the extrema of functions...today we're learning the second way:

-Introduction to concavity

-The Second Derivative Test

-Points of inflection

If we have enough time, we'll take on -challenging AoPS optimization problems, which essentially constitutes some practice of how calculus is actually used in real life: maximizing the profit of your company given limited resources or some other constraint, etc. (Sneak peak: the technique most often employed in order to deal with optimization problems in multivariable functions is called the method of Lagrange multipliers in case you're interested in finding out more about it).

Session 21

#### Contextual applications of differentiation

(Session time: 1h15m)

Ok, well I guess I half lied...there are more proofs (did you read this before we had Session 10? haha), but I promise there are just a few. We'll be differentiating all the inverse trigonometric functions and proving the legitimacy of a new rule for differentiation: Chain Rule! Of course, after proving these (25m) we'll spend some time applying what we learnt (50m).

Session 22

#### Meetup

(Session time: 1h30m)

Today we'll be going over some problems relating to what we've covered so far, and after we'll complete the Unit Test for "Contextual applications of differentiation" on Khan Academy together.

Since today we'll be wrapping up our entire Derivatives unit, if you have any questions on anything covered in the three unit tests we've gone over, I'm willing to stay a bit longer to help.

Remember to think of questions or bring up problems you need help with for this session--that's what it's for!

Session 23

#### Integration and accumulation of change

(Session time: `1h10m)

Our third and last unit: integration!

Today we'll be going over an

-Introduction to accumulation of change

-Left and right Riemann sums; over and under-approximation

Session 24

#### Integration and accumulation of change

(Session time: 60m)

Continuing what we saw last session, we're going to talk about more types of sums such as

-Midpoint and trapezoidal sums,

then rigorously define the

-Definite integral using as the limit of Riemann sums.

This is going to be a really important definition, so make sure to pay attention (almost as important as the definition of a derivative using limits)

Session 25

#### Review

Session time: 1h30m

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.

After going through all this (45min), we'll go over an integral technique *pun intended to find antiderivatives:

- U-substitution

Session 26

#### Meetup

Session time: 1h30m

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.

After going through all this (45min), we'll go over an integral technique *pun intended to find antiderivatives:

- U-substitution

Session 27

#### Meetup

(Session time: 1h30m)

Another integration technique as worthwhile knowing as u-substitution:

-Integration by parts.

We will use both techniques to

-Derive the antiderivatives of all the previously mentioned functions

Session 28

#### Meetup

(Session time: 1h30m)

Today we'll work on practicing everything we've covered this unit, as well as on a short Q&A as always, for the first 30 minutes. After, as always, we'll go over the corresponding Unit Test.

Session 29

#### Meetup

(Session time: 90m)

I think what's below speaks for itself:

Proof of The Fundamental Theorem of Calculus:

-Definition of antiderivative and indefinite integral

-Developing some methods to evaluate definite integrals, such as the Sum Rule for integrals and the

-The Reverse Power Rule