Calculus 2 Lecture 6.1: The Natural Log Function

Calculus 2 Lecture 6.1: The Natural Log Function

[jordantredway]

I have to tell you, Professor, as an Electrical Engineer, I am finding myself at a point in my career where I am showing Calculus to technicians. These people get two year trade degrees, they can use CAD tools, they are good programmers, but they lack Calculus and University level Physics. I’m no teacher, but your videos have inspired some of these young people to advance their educations. You should seriously stop for a moment and consider the effect you have. Because you are willing to share your knowledge beyond your own classroom, people are becoming fascinated with the single subject, that one topic that scared them, that caused them to settle for less than they wanted at some point. Your willingness to share freely with the world is inspiring people you’ve never met to learn the dreaded Calculus. (It’s personally my favorite subject, I double majored in E. Engineering and Mathematics, but I’m incapable of teaching. I can’t simplify or explain things to people at their level.) I just think that someone who does that sort of thing should be applauded, and I hope you continue to do so for years to come.

[mlondolozimtho]

Hi Professor Leonard, I hope you get to read this message. My name is Mlondi and I am all the way from South Africa, Johannesburg. You literally helped me to pass my calculus (from single integrals to triple integrals) after I struggled with it for a good 4 years until I saw your videos. Finally after 7 years I graduated as an Electrical and Electronic Engineer. I am currently working at my dream company as a graduate intern. You don't know how much grateful I am because of your videos. MAY THE GOOD LORD ABUNDANTLY BLESS YOU AND YOUR YOUTUBE CHANNEL GROW EXPONENTIALLY PROF!

[darthboganis]

Properties of Natural log 0:00
w/ Examples 4:30
Graph of ln (x) 23:00
Integral/anti-derivative review 28:30
Integral of 1/x 37:20 and 46:30
ln (x) = integral of 1/x 44:00
w/ examples 47:30
Chain Rule/ ln (x) 51:30
w/ Examples 54:10
Logarithmic Differentiation 1:27:30
> taking the natural log of both sides >> implicit differentiation essentially. the ln (y) involves the chain rule, because it's a function of x. d/dx [ln(y)] = 1/y * dydx (1:34:00) plug in y to express function explicitly
Another Implicit differentiation example 1:38:40

Integrals of ln 1:43:00 using substitution method for simplification
> Integrals work when they're in the right format. Substitution makes it possible to integrate 'weird' looking functions. Otherwise, consider algebraic manipulation.
Trigonometry Integrals 1:54:00

> The trick to solving these kinds of integrals is knowing how to manipulate them using the trig identities. rewrite them that is
Trig Integral examples 2:12:00
>double-angle identity example [sin (2x)] 2:19:00
> dropping the absolute value/why? 2:21:00


The derivative of ln (x) = 1/x, and the integral of 1/x = ln |x| - absolute notation because it can't be negative.

[teddylabarca]

Hi Professor Leonard. I know you probably won't see this. I would just like to say thanks for all of the videos you posted. They really helped me to succeed in calculus. I recently took calc 1 and 2 in my first year of college, and earned an A in both classes. I watched your videos before each semester started, as I thought it would be a good way of preparing for each class. I would highly recommend other calculus students to watch your videos if they need help. Learned a lot from watching them!

[AtticusEluih]

Raising the bar in the classroom and the gym.. @Professor Leonard

[MrCompenso]

My current math teacher makes me worried.  You restore my confidence.  You have no idea what that means to me.

[richarddunn]

8 years later and your videos are still helping people. I'm an adult returning to college to complete my engineering bachelors degree. Your videos have gotten me through ALL my math courses. I can't thank you

[ScopedKilz]

That guy who let out the gasp at 0:04. I felt that.

I took calc II a few years back. Just wanted to come back here to watch some old stuff for fun (now that I'm a software engineer). It's a tough class no doubt but just keep working at it and watching Prof Leonard. His videos helped me out so much when I was taking it. Best of luck!

[scashion]

The way this guy talks and explains things is so personable and uplifting, while also helping the material's ability to be understood. 10/10 professor W

[imranrashik]

just made an A in honors calculus 2! this is a message to all of you that started with calc 2 ..prof loenard is the best. I made a 67 on midterm 1 and then after watching his videos, a 96 on midterm 2 and finally an A on the course ! I feel blessed to have worked with him last semester. Thanks for everything professor :)

[aisafetyawareness]

It's insane how good he is at teaching. Goes through all the fundamentals.

[jhes]

I like how you put the long video to cover the whole lecture for the material. It doesn't mater if it will be 2 or 3 hours, this make it much easier to learn from start till the end.

Thank you for the videos.

[ViaVanille]

Your students are so blessed to have you as their instructor. Very friendly, very entertaining, and you truly do explain everything so well. Thank you for your lessons. Helping me prep for my exam in two days!

[MrBashir]

Dear Prof,
I must let you know that i have combed the internet inside out, searching for videos that even a person with zero maths educ can be able to follow, at this point i must confess that to the best of my knowledge there is no one with exceptional instructional delivery skills such as you. Your lessons are simply captivating to follow, as a maths novice PhD student (enrolled in a quantitative Prog) i have sat for hours and days following your classes step by step, and it's been very very helpful to me in following my first year doctoral courses like maths for SS, statistics, Multiv Ana, Game Theory etc. Words cannot describe the positive impact you have had on countless other people around world (more so all for free). Otherwise, without people like you, we from the developing countries wouldn't have had this opportunity. In short you are God sent and only he can compensate you.
With highest regards
Bashir

[DrumsTheWord]

Here I am starting Calc 2 after completing Calc 1 from your channel. Thanks so much for these videos!

[xgerbil]

You are a great teacher. Instead of assuming everyone knows the material, you give everyone a refresher. I seriously can't thank you enough for uploading these videos.

[dankang]

This man is my math savior!!! The Superman pic is pretty appropriate I'd say. I can really tell he has a passion for helping students understand the logic behind each concept...I wish more instructors were like him. Keep doin what you're doin Professor Leonard.

[Schrodinger]

I collected all the material, lectures, books & homework to sync with his videos. The full guide to Calc II Prof. Leonard Edition. He used Single Variable Calculus by Tan 1st edition.

Course:

Chapter 6: Transcendental Functions
o Lecture 6.1: The Natural Log Function
o Lecture 6.2: Derivatives of Inverse Functions
o Lecture 6.3: Derivatives and Integrals of Exponential Functions
o Lecture 6.4: Derivatives and Integrals of General Exponential Functions
o Lecture 6.5: Calculus of Inverse Trigonometric Functions
o Lecture 6.6: A Discussion of Hyperbolic Functions
o Lecture 6.7: Evaluating Limits of Indeterminate Forms

Chapter 7: Techniques of Integration
o Lecture 7.1: Integration By Parts
o Lecture 7.2: Techniques For Trigonometric Integrals
o Lecture 7.3: Integrals By Trigonometric Substitution
o Lecture 7.4: Integration By Partial Fractions
o Lecture 7.6: Improper Integrals

Chapter 8: First-order Differential Equations
o Lecture 8.1: Solving First Order Differential Equations By Separation of Variables

Chapter 9: Infinite Sequences and Series
o Lecture 9.1: Convergence and Divergence of Sequences
o Lecture 9.2: Series, Geometric Series, Harmonic Series, and Divergence Test
o Lecture 9.3: Using the Integral Test for Convergence/Divergence of Series, P-Series
o Lecture 9.4: The Comparison Test for Series and The Limit Comparison Test
o Lecture 9.5: Showing Convergence With the Alternating Series Test, Finding Error of Sums
o Lecture 9.6: Absolute Convergence, Ratio Test and Root Test For Series
o Lecture 9.7: Power Series, Calculus of Power Series, Ratio Test for Int. of Convergence
o Lecture 9.8: Representation of Functions by Taylor Series and Maclauren Series
o Lecture 9.9: Approximation of Functions by Taylor Polynomials

Chapter 10: Parametric Equations and Polar Coordinates
o Lecture 10.2: Introduction to Parametric Equations
o Lecture 10.3: Calculus of Parametric Equations
o Lecture 10.4: Using Polar Coordinates and Polar Equations
o Lecture 10.5: Calculus of Polar Equations
o Lecture 4.6: Numerical Integration With Trapezoidal and Simpson's Rule

Homework problems:

6.1 pg. 528-529 #1, 2, 5-9, 11-14, 27-39, 43, 44, 65-68, 71-82
6.2 pg. 538-540 #1-14, 21-25, 29-33, 47-52
6.3 pg. 549-553 #1-9, 15-39, 82-98
6.4 pg. 565 #1-9, 21-50
6.5 pg. 575-576 #1-22, 33-54, 65-84
6.6 pg. 587 #1-3, 7-15, 19-34, 38-52, 55-62
6.7 pg. 598 #1-19, 23-31, 35-45, 49-55
7.1 pg. 613-614 #1-42
7.2 pg. 623 #1-48
7.3 pg. 631 #1-31
7.4 pg. 642-643 #1-43
7.6 pg. 665 #7-47
8.1 pg. 687-688 #1, 2, 9-14, 19-24, 35, 40, 41
9.1 pg. 743-744 #1-9, 13-39, 51-54
9.2 pg. 754-755 #1-5, 7-9, 11-20, 29-40, 44-48
9.3 pg. 760 #1-5, 9-24, 26-29, 32
9.4 pg. 767-768 #1-10, 13-22, 25-34
9.5 pg. 773-774 #1-16, 21-22, 27-28
9.6 pg. 783 #1-30
9.7 pg. 792 #1-26, 39
9.8 pg. 805-806 #1-6, 8, 11-22, 26-28, 47
9.9 pg. 820-821 #3-11, 15, 17-22
10.2 pg. 855 #1-20
10.3 pg. 863-865 #1-5, 7-9, 11, 13-15, 17-21, 31-33, 52-53, 58-59
10.4 pg. 877 #1-16, 25-29, 33-36, 43-48, 65-68
10.5 pg. 885-887 #1-5, 13-16, 29-32, 35-38, 41-44, 47, 56
4.6 pg. 429 #1-10, 15, 16

Week - Video's:

1 6.1 – 6.2
2 6.3 – 6.4
3 6.5
4 6.6
5 6.7
6 7.1 – 7.2
7 7.3
8 7.4
9 7.6
10 8.1
11 9.1 – 9.2
12 9.3 – 9.4
13 9.5 – 9.6
14 9.7 – 9.8
15 9.9
16 10.2 – 10.4
17 10.5, 4.6

[Adrian-czob]

"Show hands if you feel ok with that one"
me behind the screen = 🤚

[mohamedabdelaal]

The way you throw in small motivations/ compliments like "if you got to here, you know what you're doing" Really motivates me!!!! You're the best teacher ever!!!