Distance from a Point to a Plane using the Dot Product https://youtu.be/-V6F0OpS5M8

In this video I go over a quick recap on the formula for determining the shortest distance between a point and a plane in 3D. The formula is just the projection of any position vector from the plane to the point onto the normal vector of the plane, and involves their dot product.

#math #science #geometry #STEM #education

Distance Between Intersecting Lines, Parallel Lines, and Skew Lines https://youtu.be/hWqpXXNZK0U

In this video I discuss how to determine the distance between two lines for each of the 3 possible cases: intersecting lines, parallel lines, and skew lines (which don't intersect and are not parallel).

#math #science #geometry #education #STEM

Traces of a Surface: Review and Example on an Ellipsoid whose Traces are Ellipses https://youtu.be/w72pnqHHbAA

In this video I review traces of a surface, which are curves of intersection of a surface with planes parallel to the coordinate (x, y, z) planes.

#math #science #geometry #STEM #education

Equations of the Six Types of Quadric Surfaces https://youtu.be/0pCWT-FbmuY

In this video I review the standard form equations of the 6 types of quadric 3D surfaces: ellipsoid, cone, elliptic paraboloid, hyperboloid of one sheet, hyperbolic paraboloid, and hyperboloid of two sheets.

#math #science #geometry #education #STEM

Cell Structure and Organelles of Animal and Plant Cells https://youtu.be/_0Qofjy4m0c

In this video I go over the typical cell structures of animal and plant cells as well as the many organelles that comprise them. Both cells contain a nucleus with most of the organism's genome, besides the mitochondrial DNA in animals and plastid DNA in plants.

#science #biology #STEM #education #animals

Cell Cycle: Mitosis, Meiosis, and Binary Fission https://youtu.be/dsxUZtwVPhM

In this video, I explain the cell cycle, the series of events a cell undergoes to divide into two (or four) daughter cells. In eukaryotic cells (such as animal and plant cells), there are two main types of cell division: mitosis and meiosis.

#biology #science #chemistry #education #STEM

Recap on Power Series, Taylor Polynomials, and Taylor's Inequality https://youtu.be/xajyzNArX68

In this video I go over a quick recap on power series, which is a specific type of infinite series whose terms are powers of x multiplied by coefficients. A Taylor series is a specific type of power series whose coefficients are determined from the coefficients of a function.

#math #science #education #calculus #STEM

Approximating Functions by Taylor Polynomials https://youtu.be/woT9NoaFM_Y

In this video, I demonstrate how Taylor polynomials can be used to approximate functions and how to evaluate the accuracy of these approximations by examining the absolute magnitude of the error.

#math #science #calculus #education #STEM

Example 1: Approximating Cube Root Function by a 2nd Degree Taylor Polynomial https://youtu.be/0616gOyjbyQ

In this video I approximate the cube root function x^(1/3) by using a 2nd degree Taylor polynomial at a = 8. The first step is to obtain the first 2 derivatives of f at x = 8, and then the third derivative to be used for the error estimation.

#math #science #calculus #education #STEM

Exercise 1: Taylor Derivatives Equal the Function's Derivatives at x = a https://youtu.be/UqWf2PE1OeE

In this video I prove that the derivatives of a Taylor Polynomial are equal to the derivatives of the corresponding function used in the approximation when x = a. This is because any derivative at x = a of any Taylor polynomial cancels out all the terms (since a - a = 0) except the first term, which is just the corresponding derivative of f at x = a.

#math #science #education #STEM #calculus

Exercise 5: Approximating the Period of a Pendulum using the Binomial and Geometric Series https://youtu.be/_6XvO2pRMW0 - playlist - notes

In this video I go over the formula for the period (time it takes to go back and forth) of a pendulum and approximate it using the binomial and geometric series.

#math #science #education #physics #STEM

Problems Plus 1: Tightly Packed Balls in a Box (and Tightly Packed Circles in a Square) https://youtu.be/Rs8YkurKjCM - notes - playlist

In this video I determine the radius of 9 balls that are tightly packed inside a box. I first solve the 2D case by solving for the radius of 5 circles tightly packed inside a square.

#math #science #education #geometry #STEM

Problems Plus 2: Volume Around a Box (and Area Around a Rectangle) https://youtu.be/bNZyuUB0D4w - notes - playlist

In this video I determine the volume of a region whose distance is at most 1 from a box. To understand how to solve this, I solve the 2D case first and solve the area around a rectangle.

#math #science #education #geometry #STEM

Evolutionary Biology: Phylogenies, Homology, and Taxonomy https://youtu.be/apZQ35bMOcE - notes - playlist

In this video I go over phylogenies, which represent the evolutionary history of a group of organisms or their genes.

#science #biology #evolution #STEM #education

Evolutionary Biology: Speciation, Reproductive Isolation, and Hybrids https://youtu.be/u_ygO92LgmY - notes - playlist

In this video, I explain speciation, the process by which one lineage splits into two independent lineages that evolve separately. For speciation to occur, reproductive isolation is required, which can result from physical barriers (like geographic separation) or genetic incompatibilities.

#science #biology #evolution #STEM #education

Biological Diversity: Bacteria, Archaea, and Eutrophication https://youtu.be/KOgugykmT6Q - notes - playlist

In this video I go over bacteria and archaea which comprise two large and separate domains of prokaryotic microorganisms.

#science #biology #bacteria #education #STEM

Biological Diversity: Protists https://youtu.be/3GV0vZYZ9tM - notes - playlist

In this video I discuss protists, which are eukaryotic organisms that are not animals, plants, or fungi.

#science #biology #protists #education #STEM