Calculus • Series
Comprehensive Calculus: Limits, Derivatives, and Integrals
PL
JG
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
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 - July 24
Learners
30 / 40
Total Sessions
39
About the Tutor
PL
JG
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 gotten 5s on AP Spanish Lit and AP English Lang, and I've taken Multivariable Calc and Linear Algebra (which I hope to tutor soon!) I'm currently taking AP Physics C and Real Analysis.
View Jose Roberto Cossich G's Profile
Upcoming Sessions
0
Past Sessions
39
26
Mar
Lc
Session 1
Limits and continuity
Hey everyone! (Session time: 1h30m)
Today we'll be starting off with an introduction to the series, so we'll have a little orientation session to break the ice and get familiarized with how this series works. This should only take 20-30 minutes, so we'll use the remaining time to start off our discussion on limits.
Topics covered today:
- Introduction to limits
- One and two-sided limits
- Creating tables to approximate limits; approximating limits using tables
27
Mar
Lc
Session 2
Limits and continuity
(Session time: 60m)
Hey guys, today we'll be covering the following topics:
- Limits of composite functions
- Evaluating limits through direct substitution
28
Mar
Lc
Session 3
Limits and continuity
(Session time: 1h30m)
Hey guys, today we'll be covering the following topics:
-Limits of trigonometric and piecewise functions
-Evaluating limits by rationalizing and by using trigonometric identities
-Squeeze theorem and applications
30
Mar
Lc
Session 4
Limits and continuity
(Session time: 1h30m)
Hey guys, today we'll be covering the following topics:
-Limits of trigonometric and piecewise functions
-Evaluating limits by rationalizing and by using trigonometric identities
-Squeeze theorem and applications
4
Apr
Lc
Session 5
Limits and continuity
(Session time: 1h30m)
Hey guys, today we'll be covering the following topics:
-Introduction to continuity and types of discontinuities
-Continuity at a point and over an interval
-Removing discontinuities
Today we'll be practicing everything we've covered so far, and leave the last 30 minutes to cover
-Limits at infinity and limits at infinity of quotients
10
Apr
Lc
Session 6
Limits and continuity
Today we'll be wrapping up our unit by covering
-The Intermediate Value Theorem
-The formal epsilon-delta definition of a limit
(+ Links to epsilon-delta exercises)
15
Apr
R
Session 7
Review
(Session time: 2h)
Today's a review session! So it's going to be structured a bit differently. You can stay for the sections you find most useful to you.
FYI:
-20min Q&A on everything we've covered so far
-60min to work through the unit test for Limits and Continuity on Khan Academy
-30min to work through challenging problems AoPS problems.
16
Apr
Lc
Session 8
Limits and continuity
(Session time: 2h)
New content:
-Limits at infinity and limits at infinity of quotients
-The Intermediate Value Theorem
-Q&A for epsilon-delta definition of a limit and proofs of limit properties, as well as Intermediate Value Theorem and Squeeze Theorem
-Q&A on everything we've covered so far
-60min to work through the unit test for Limits and Continuity on Khan Academy
17
Apr
Dr
Session 9
Differentiation: definition and basic derivative rules
(Session time: 1h15m)
Hey guys! I'm really excited to start to start working on second unit: Derivatives!
Today we'll cover the following topics, and wrap up with a nice game:
-Introduction to derivatives and notation
-Average rate of change and the secant line
-Estimating derivatives graphically and algebraically
18
Apr
Dr
Session 10
Differentiation: definition and basic derivative rules
(Session time: 2h)
Proofs, proofs, proofs!
Today we'll be getting to one of my favorite parts of this unit, proving the Sum Rule, Power Rule, Product Rule, and Quotient Rule--all different techniques which allow us to differentiate functions! That should take us around an hour or so, so we'll use the remaining hour to practice these new and *extremely important techniques.
20
Apr
Dr
Session 11
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!
22
Apr
R
Session 12
Review
(Session time: 1h10m)
Today we'll be practicing all the differentiation rules we've learnt so far, first through some practice problems and then by completing the Unit Test together for the "Differentiation: definition and basic derivative rules" unit. If we have time, we'll practice differentiating inverse trigonometric functions and using chain rule some more.
21
May
Df
Session 13
Differentiation: composite, implicit, and inverse functions
(Session time: 1h20m)
These are other favorite topics of mine, because whoever thought of them must have been so clever:
-Related rates problems
-Local linearity and linear approximations to functions; deriving equation of the tangent line of a function at a point
Df
Session 14
Differentiation: composite, implicit, and inverse functions
(Session time: 1h30m)
Now that we've learnt all the basic theory, we can apply our knowledge of derivatives by dealing with
-Second derivatives
-Implicit differentiation
-Position, velocity, and acceleration problems
I have a feeling this will be particularly interesting to those interested in Physics...when we learnt antidifferentiation (integration) we'll be proving some basic projectile motion formulas!
22
May
Cd
Session 15
Contextual applications of differentiation
(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.
27
May
Cd
Session 16
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!
29
May
Dr
Session 17
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!
30
May
Cd
Session 18
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
4
Jun
Cd
Session 19
Contextual applications of differentiation
-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
Additionally, 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.
We'll be differentiating all the inverse trigonometric
5
Jun
Cd
Session 20
Contextual applications of differentiation
- Continuing last session
6
Jun
Cd
Session 21
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).
8
Jun
OT
Session 22
Other Topics
(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).
11
Jun
OT
Session 23
Other Topics
(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).
12
Jun
OT
Session 24
Other Topics
(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 time: 1h30m)
Now that we've learnt all the basic theory, we can apply our knowledge of derivatives by dealing with
-Second derivatives
-Implicit differentiation
-Position, velocity, and acceleration problems
I have a feeling this will be particularly interesting to those interested in Physics...when we learnt antidifferentiation (integration) we'll be proving some basic projectile motion formulas!
15
Jun
OT
Session 25
Other Topics
(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).
19
Jun
OT
Session 26
Other Topics
(Session time: 1h30m)
Today we'll be going over the derivatives of inverse functions in general as well as the derivatives of inverse trigonometric functions arcsin(x) and arccos(x) (I will consider including the other four). We will also apply the chain rule to find the derivatives of a^x and log_a(x).
OT
Session 27
Other Topics
(Session time: 3h)
Today we'll be going over some problems relating to what we've covered so far, and after we'll complete the Unit Tests for "Differentiation: definition and basic derivative rules," "Contextual applications of differentiation," and "Differentiation: composite, implicit, and inverse functions" 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!
21
Jun
Ic
Session 28
Integration and accumulation of change
(Session time: 1h30m)
Today we'll be going over the derivatives of inverse functions in general as well as the derivatives of inverse trigonometric functions arcsin(x) and arccos(x) (I will consider including the other four). We will also apply the chain rule to find the derivatives of a^x and log_a(x).
23
Jun
Ic
Session 29
Integration and accumulation of change
(Session time: 3h)
Today we'll be going over some problems relating to what we've covered so far, and after we'll complete the Unit Tests for "Differentiation: definition and basic derivative rules," "Contextual applications of differentiation," and "Differentiation: composite, implicit, and inverse functions" 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!
25
Jun
Ic
Session 30
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)
28
Jun
Ic
Session 31
Integration and accumulation of change
(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
29
Jun
Ic
Session 32
Integration and accumulation of change
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
2
Jul
Cd
Session 33
Contextual applications of differentiation
- Going over Unit 5 unit test and Unit 4 content as review
Ic
Session 34
Integration and accumulation of change
- Riemann sums in summation notation\definite integral express as a limit
-Quiz 1, 2, and 3
4
Jul
Ic
Session 35
Integration and accumulation of change
- Up to Quiz 4 (common antiderivatives)
-U-substitution
5
Jul
Ic
Session 36
Integration and accumulation of change
-Integration by parts
-Integration by partial fractions
6
Jul
Ic
Session 37
Integration and accumulation of change
-Average value of a function
-Area between curves
-Distance and velocity from acceleration with initial conditions
-Solutions to exponential differential equations
8
Jul
R
Session 38
Review
Our last session, covering methods of integration
23
Jul
Ic
Session 39
Integration and accumulation of change
Our last session, covering methods of integration