Approximating Optical Refraction using Taylor Polynomials https://youtu.be/Y80dBUrTCh8

In this video, I demonstrate how Taylor polynomials are used in optical physics to simplify the complex refraction equation.

#math #science #physics #education #optics

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 2: Deriving the 1st Order and 3rd Order Optics Formulas https://youtu.be/leZ0QK_n07k - notes

In this video, I derive the first-order and third-order optics formulas by approximating the refraction formula for light using first and third-order Taylor polynomials, respectively.

#math #physics #optics #education #science

Exercise 3: Approximating Electric Field of Electric Dipoles using Binomial Series https://youtu.be/RSkkBiOjXnw - playlist - notes

In this video, I use the binomial series approximation to demonstrate that the electric field (E) of an electric dipole (two equal but opposite charges separated by a small distance) is proportional to 1 over the distance cubed when measured from a far distance.

#math #physics #science #education #calculus

Exercise 4: Approximating the Velocity of a Shallow Water Wave using Maclaurin Series https://youtu.be/Q_IXu-V8vng - playlist - notes

In this video I go over the water wave velocity equation and approximate it when the waves are deep and when the waves are shallow.

#math #science #physics #water #education

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

Exercise 6: Surveying: Elevation Correction due to Earth's Curvature using Taylor Polynomials https://youtu.be/T0nNsO_qA84 - playlist - notes - calculator

In this video I derive the elevation correction required when surveying long distances across a highway due to Earth's curvature, and also derive a Taylor polynomial approximation for the correction. For an 100 km long highway, the Taylor polynomial is accurate to within an amazing 0.00808 mm!

#math #science #education #physics #Earth

Vectors and the Geometry of Space: Problems Plus https://youtu.be/4x6uh1zQ-M0 - notes - playlist

In this video I go over the end-of-chapter Problems Plus advanced math problems for the Vectors and Geometry of Space chapter. There are 8 problems in total and cover 3D geometry, surfaces, vectors, vector projections, and applications to physics and airplane dynamics. It's 4 hours of advanced math so buckle up!

#math #science #physics #geometry #calculus

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

Problems Plus 3: Line and Surface of Two Intersecting Planes https://youtu.be/jj3wYCYhhq8 - notes - playlist

In this video I determine the symmetric equations of a line at the intersection of two planes. I then find out that when the equations of the plane vary, the line sweeps out a circular hyperboloid of one sheet. Finally, I calculate the volume of this surface between two planes.

#math #science #education #geometry #vectors

Problems Plus 4: Plane Flying in Wind https://youtu.be/Up-l6Nt_w08 - notes - playlist

In this video I calculate the required direction for a plane to fly accounting for the wind velocity such that the ground velocity is due North.

#math #science #physics #education #pilot

Problems Plus 5: Infinite Sum of Vector Projections is a Geometric Series https://youtu.be/rKt2qVLCMiM - notes - playlist

In this video I find the sum of an infinite series of consecutive magnitudes of vector projections and show it turns out to be just a geometric series.

#math #science #education #geometry #vectors

Problems Plus 6: Largest Sphere in a Sphere https://youtu.be/uefSE-rXFw4 - notes - playlist

In this video I find the largest sphere that passes through a given point while every point inside is also inside a larger sphere.

#math #science #geometry #education #GeoGebra

Problems Plus 7: The Physics of a Block on an Inclined Plane https://youtu.be/0ZJPy-Egfpk - notes - playlist

In this video I go over the physics of a solid block on an inclined plane in terms of the sliding friction, weight, normal force, and applied horizontal force.

#math #science #geometry #education #physics

Problems Plus 8: Volume of a Solid from its Shadows https://youtu.be/S-MGdEU9DF0 - notes - playlist

In this video I obtain the volume of the largest solid whose shadows are a square of height 1, a circle, and an isosceles triangle.

#math #science #geometry #education #physics

An Elusive Limit Questions 1, 2: Limit Depends on Calculator Precision https://youtu.be/ULTOgc1cuAI - notes - playlist

In this video I go over a very interesting trigonometric (and inverse trigonometric) function whose limit as x approaches 0 is unclear and gives different values depending on the calculator you use.

#math #science #education #calculus #Desmos

An Elusive Limit Question 3: Using L'Hospital's Rule 7 Times in a Row (Using Wolfram Alpha) https://youtu.be/LRLahyO4SVk - notes - playlist

In this video I go over the previous difficult trigonometric (and inverse trigonometric) function and this some apply L'Hospital's Rule to show that its limit as x approaches 0 is equal to 1.

#math #science #education #calculus #WolframAlpha

An Elusive Limit Question 4-5: Evaluating the Limit using Taylor Series (and Wolfram Alpha) https://youtu.be/GEfzOVaPwAE - notes - playlist

In this video I go over the earlier trigonometric (and inverse trigonometric) limit and solve it by writing the numerator and denominator as Taylor series.

#math #science #education #calculus #WolframAlpha

An Elusive Limit Question 6: Subtracting Nearly Identical Numbers can BREAK Calculators! https://youtu.be/GEfzOVaPwAE - notes - playlist

In this video I explain that the reason calculators gave an unclear limit for the complicated trigonometric function is because every term was nearly identical, to the point where subtracting them resulted in a difference too small for the precision of most calculators.

#math #science #education #calculus #Desmos