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

Example 2: Approximating sin(x) with Taylor Polynomials and Calculating its Error https://youtu.be/FYD8E-kxpnk

In this video I go over a very in-depth video exploring the trigonometric sine function, its Taylor polynomials approximation, and its associated error.

#math #science #calculus #trigonometry #education

Example 3: Einstein's Theory of Special Relativity vs. Classical Theory for Kinetic Energy https://youtu.be/OQrJllLdbp0

In this video I approximate Albert Einstein's kinetic energy equation from his theory of special relativity and show that at low velocities the equation agrees with the classical Newtonian equation K = (1/2)mv^2.

#math #science #physics #Einstein #education

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