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1.[Pilot Section] If you just finished school What you know & what you will learn\ 0:00 Overview of the course 4:51 Section intro 7:11 Overview of this pilot section What you know and what you don’t know yet 13:02 What exactly is a function 20:51 Linear functions 28:04 Quadratic functions and solving quadratic equations 41:59 Factorizing polynomials using their roots & Outlook Complex numbers 56:42 Exponential function How is it defined exactly 1:09:45 Vector algebra in polar coordinate system 1:19:17 Vector rotation using matrices 1:24:57 Section outro 2.[Part 1] College-level What you are expected to know at university\ 1:25:49 Overview of the first part of this course of functions\ 1:27:11 Section intro 1:28:13 Limits Dealing with infinity 1:36:16 Limits versus function values 1:44:30 Polynomial fractions 1:53:47 Asymptotic behavior 2:01:00 Useful rules 2:02:53 Your turn! About exercises 2:04:14 Solutions of Tasks 1 & 2 - 3 & 4 - Limits of functions 2:22:25 Add on Two advanced lectures & results that we need later on 2:24:21 Squeeze theorem Limit of sin(x)x for x to 0 2:34:05 L’Hospital’s rule (L’Hôpital’s rule) 2:39:19 Section outro in one dimension\ 2:39:35 Section intro 2:42:21 What is a derivative 2:49:38 Derivative of constant, linear and quadratic functions 2:58:23 Sum rule for derivatives 3:02:25 Derivative of polynomials 3:14:57 Product rule for derivatives 3:23:56 Derivative of the 1 x function 3:27:56 Chain rule for derivatives 3:38:43 Quotient rule for derivatives 3:45:02 Derivative of the inverse function 3:49:40 Derivative of root functions 3:54:54 Derivative of power functions 3:58:44 Derivative of exponential & logarithm functions 4:04:32 Derivative of trigonometric functions Sine & Cosine 4:09:18 Outlook Derivative of trigonometric functions by series expansion 4:17:32 Higher derivatives 4:21:11 Extrema of functions & Inflection points 4:34:54 Curve sketching 4:42:35 Summary of the rules for derivatives 4:44:58 Your turn! About exercises 4:45:26 Solutions Task 1 - Derivatives 4:57:33 Solutions Tasks 2 & 3 - Derivatives 5:05:00 Add on How to calculate derivatives numerically 5:08:07 Numerics with Python Calculating derivatives in one dimension\ 5:27:23 Section intro 5:29:11 What is an integral 5:39:37 Integration by parts 5:49:34 Integration by substitution 5:59:02 Alternative way of using integration by substitution 6:05:21 Summary of integration rules 6:09:21 Integration & Limits - Improper integrals 6:15:44 Integration versus differentiation 6:20:48 Integration of cosine square & sine square 6:25:54 Exploiting symmetry 6:32:11 Solutions Task 1 - 2 - Integrals 6:51:19 Add on How to calculate integrals numerically 6:52:45 Numerics with Python Calculating integrals in Cartesian coordinates\ 7:07:53 Section intro 7:09:28 What is a vector 7:14:30 Basic vector operations 7:22:26 Dot product or scalar product 7:28:47 Cross product or vector product 7:35:59 Triple product 7:40:41 Lines in parametric form 7:47:34 Lines & Points Calculating the distance 7:58:20 Lines & Lines Identical, parallel, intersecting or skew lines 8:11:04 Planes in parametric form 8:15:58 Planes in coordinate form 8:23:33 Planes in Hesse normal form 8:32:00 Planes & Points Calculating the distance 8:40:29 Planes & Lines Included, parallel or intersecting 8:44:26 Planes & Planes Identical, parallel or intersecting line 8:51:16 Solutions Task 1 - Vectors Two points 8:55:35 Solutions Task 2 - Vectors Two lines 9:11:56 Solutions Task 3 - Vector algebra Planes 9:22:00 Solutions Task 4 - Vector identities 9:29:23 Solutions Task 5 - Vectors Distance between two lines 9:38:12 Section outro & Probability distributions\ 9:38:41 Section intro 9:39:29 Probability & Tree diagrams for coin flip experiments 9:44:44 Event & Counter event in a dice experiment 9:51:31 Expectation values for coin, dice & urn problems 10:01:26 Calculating probabilities Urn problems 10:12:51 Binomial distribution 10:19:00 Discussion of the binomial distribution 10:28:28 Normal distribution (Gaussian distribution) 10:32:39 Poisson distribution 10:35:52 Solutions Task 1 - 2 - 3 - Probabilities 10:52:57 Section outro 8.[Part 2] University-level What is new at university\ 10:53:17 Overview of the second part of this course & Series\ 10:57:54 Section intro 11:00:44 Sequences 11:10:24 Limits of sequences 11:18:14 Series - Harmonic & Geometric series 11:26:55 Series & Relation to improper integrals 11:31:49 Your turn! About exercises 11:32:38 Solutions Task 1 - Sequences 11:37:40 Solutions Task 2- Series expansion & Series representation of Exponential & Trigonometric function\ 11:43:31 Section intro 11:44:53 What is a Taylor expansion 11:52:35 Proof for Taylor series 11:58:25 Examples of Taylor series
