Calculus – Frank's World of Data Science & AI

Mathematics
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Free Full Calculus 1 College Course

Learn Calculus 1 in this full college course.

This course was created by Dr. Linda Green, a lecturer at the University of North Carolina at Chapel Hill. Check out her YouTube channel: https://www.youtube.com/channel/UCkyLJh6hQS1TlhUZxOMjTFw

Lecture Notes

Calculus 1 Corequisite Notes: http://lindagreen.web.unc.edu/files/2020/08/courseNotes_math231L_2020Fall.pdf
Calculus 1 Notes: http://lindagreen.web.unc.edu/files/2019/12/courseNotes_m231_2018_S.pdf

Course Contents:

(0:00:00) [Corequisite] Rational Expressions
(0:09:40) [Corequisite] Difference Quotient
(0:18:20) Graphs and Limits
(0:25:51) When Limits Fail to Exist
(0:31:28) Limit Laws
(0:37:07) The Squeeze Theorem
(0:42:55) Limits using Algebraic Tricks
(0:56:04) When the Limit of the Denominator is 0
(1:08:40) [Corequisite] Lines: Graphs and Equations
(1:17:09) [Corequisite] Rational Functions and Graphs
(1:30:35) Limits at Infinity and Graphs
(1:37:31) Limits at Infinity and Algebraic Tricks
(1:45:34) Continuity at a Point
(1:53:21) Continuity on Intervals
(1:59:43) Intermediate Value Theorem
(2:03:37) [Corequisite] Right Angle Trigonometry
(2:11:13) [Corequisite] Sine and Cosine of Special Angles
(2:19:16) [Corequisite] Unit Circle Definition of Sine and Cosine
(2:24:46) [Corequisite] Properties of Trig Functions
(2:35:25) [Corequisite] Graphs of Sine and Cosine
(2:41:57) [Corequisite] Graphs of Sinusoidal Functions
(2:52:10) [Corequisite] Graphs of Tan, Sec, Cot, Csc
(3:01:03) [Corequisite] Solving Basic Trig Equations
(3:08:14) Derivatives and Tangent Lines
(3:22:55) Computing Derivatives from the Definition
(3:34:02) Interpreting Derivatives
(3:42:33) Derivatives as Functions and Graphs of Derivatives
(3:56:25) Proof that Differentiable Functions are Continuous
(4:01:09) Power Rule and Other Rules for Derivatives
(4:07:42) [Corequisite] Trig Identities
(4:15:14) [Corequisite] Pythagorean Identities
(4:20:35) [Corequisite] Angle Sum and Difference Formulas
(4:28:31) [Corequisite] Double Angle Formulas
(4:36:01) Higher Order Derivatives and Notation
(4:39:22) Derivative of e^x
(4:46:52) Proof of the Power Rule and Other Derivative Rules
(4:56:31) Product Rule and Quotient Rule
(5:02:09) Proof of Product Rule and Quotient Rule
(5:10:40) Special Trigonometric Limits
(5:17:31) [Corequisite] Composition of Functions
(5:29:54) [Corequisite] Solving Rational Equations
(5:40:02) Derivatives of Trig Functions
(5:46:23) Proof of Trigonometric Limits and Derivatives
(5:54:38) Rectilinear Motion
(6:11:41) Marginal Cost
(6:16:51) [Corequisite] Logarithms: Introduction
(6:25:32) [Corequisite] Log Functions and Their Graphs
(6:36:17) [Corequisite] Combining Logs and Exponents
(6:40:55) [Corequisite] Log Rules
(6:49:27) The Chain Rule
(6:58:44) More Chain Rule Examples and Justification
(7:07:43) Justification of the Chain Rule
(7:10:00) Implicit Differentiation
(7:20:28) Derivatives of Exponential Functions
(7:25:38) Derivatives of Log Functions
(7:29:38) Logarithmic Differentiation
(7:37:08) [Corequisite] Inverse Functions
(7:51:22) Inverse Trig Functions
(8:00:56) Derivatives of Inverse Trigonometric Functions
(8:12:11) Related Rates – Distances
(8:17:55) Related Rates – Volume and Flow
(8:22:21) Related Rates – Angle and Rotation
(8:28:20) [Corequisite] Solving Right Triangles
(8:34:54) Maximums and Minimums
(8:46:18) First Derivative Test and Second Derivative Test
(8:51:37) Extreme Value Examples
(9:01:33) Mean Value Theorem
(9:09:09) Proof of Mean Value Theorem
(0:14:59) [Corequisite] Solving Right Triangles
(9:25:20) Derivatives and the Shape of the Graph
(9:33:31) Linear Approximation
(9:48:28) The Differential
(9:59:11) L’Hospital’s Rule
(10:06:27) L’Hospital’s Rule on Other Indeterminate Forms
(10:16:13) Newtons Method
(10:27:45) Antiderivatives
(10:33:24) Finding Antiderivatives Using Initial Conditions
(10:41:59) Any Two Antiderivatives Differ by a Constant
(10:45:19) Summation Notation
(10:49:12) Approximating Area
(11:04:22) The Fundamental Theorem of Calculus, Part 1
(11:15:02) The Fundamental Theorem of Calculus, Part 2
(11:22:17) Proof of the Fundamental Theorem of Calculus
(11:29:18) The Substitution Method
(11:38:07) Why U-Substitution Works
(11:40:23) Average Value of a Function
(11:47:57) Proof of the Mean Value Theorem for Integrals

Visualizing Deep Learning

vcubingx provides a visual introduction to the structure of an artificial neural network.

The Neural Network, A Visual Introduction | Visualizing Deep Learning, Chapter 1

0:00 Intro
1:55 One input Perceptron
3:30 Two input Perceptron
4:40 Three input Perceptron
5:17 Activation Functions
6:58 Neural Network
9:45 Visualizing 2-2-2 Network
10:59 Visualizing 2-3-2 Network
12:33 Classification
13:05 Outro

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Calculus Rhapsody

In honor or Pi Day, here is Calculus Rhapsody.

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How Decoherence Splits The Quantum Multiverse

Why is it that we can see these multiple histories play out on the quantum scale, and why do lose sight of them on our macroscopic scale?

Many physicists believe that the answer lies in a process known as quantum decoherence.

Does conscious observation of a quantum system cause the wavefunction to collapse? The upshot is that more and more physicists think that consciousness – and even measurement – doesn’t directly cause wavefunction collapse.

In fact probably there IS no clear Heisenberg cut. The collapse itself may be an illusion, and the alternate histories that the wavefunction represents may continue forever. The question then becomes: why is it that we can see these multiple histories play out on the quantum scale, and why do lose sight of them on our macroscopic scale? Many physicists believe that the answer lies in a process known as quantum decoherence.

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Does Consciousness Influence Quantum Mechanics?

It’s not surprising that the profound weirdness of the quantum world has inspired some outlandish explanations – nor that these have strayed into the realm of what we might call mysticism.

One particularly pervasive notion is the idea that consciousness can directly influence quantum systems – and so influence reality.

PBS Space Time examines where this idea comes from, and whether quantum theory really supports it.

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Grant Sanderson on the Beauty of Mathematics

Lex Fridman interviews Grant Sanderson is a math educator and creator of 3Blue1Brown, a popular YouTube channel that uses programmatically-animated visualizations to explain concepts in linear algebra, calculus, and other fields of mathematics.

OUTLINE:

0:00 – Introduction
1:56 – What kind of math would aliens have?
3:48 – Euler’s identity and the least favorite piece of notation
10:31 – Is math discovered or invented?
14:30 – Difference between physics and math
17:24 – Why is reality compressible into simple equations?
21:44 – Are we living in a simulation?
26:27 – Infinity and abstractions
35:48 – Most beautiful idea in mathematics
41:32 – Favorite video to create
45:04 – Video creation process
50:04 – Euler identity
51:47 – Mortality and meaning
55:16 – How do you know when a video is done?
56:18 – What is the best way to learn math for beginners?
59:17 – Happy moment

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Gilbert Strang on Linear Algebra, Deep Learning, Teaching, and MIT OpenCourseWare

Lex Fridman interviews Gilbert Strang on Linear Algebra, Deep Learning, Teaching, and MIT OpenCourseWare.

Gilbert Strang is a professor of mathematics at MIT and perhaps one of the most famous and impactful teachers of math in the world. His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times. This conversation is part of the Artificial Intelligence podcast.

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Understand Calculus in 10 Minutes

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Lambda Calculus

Whether you realize it or not, lambda calculus has already impacted your world as a data scientist or a developer. If you’ve played around in functional programming languages like Haskell or F#, then you are familiar some of the same ideas. In fact, AWS’s serverless product is named Lambda after this branch of mathematics.

Watch this video to learn about lambda calculus.

Tag: Calculus