Remainder Estimate for the Integral Test + Example - Hive - YouTube - Notes - Playlist - Sequences and Series

In this video, I approximate the accuracy of the partial sums of a series that converges via the Integral Test by estimating the size of the remainder. This is done by comparing the areas of the rectangles below the curve whose function corresponds to the terms of the series, but doing so for all the terms after the n-th partial sum. The areas of the rectangles will be greater than the improper integral starting from n and less than the improper integral starting from n + 1; this is the Remainder Estimate for the Integral Test!

I illustrate this with an example on the series with terms 1/n³ and show that with 32 terms in the partial sum, the accuracy is within 0.0005 of the actual sum of the infinite series.

#math #calculus #series #remainder #education

Estimating Sum of Series as the Midpoint is a much better Approximation than the Partial Sum - Hive - YouTube - Notes - Playlist - Sequences and Series

In this video, I show that modifying our earlier Theorem 2 remainder estimate by adding the partial sum to both sides of the inequality provides a much better approximation than simply using the partial sum to approximate the sum of the series. This is because we now obtain an interval rather than just a simple sum. Taking the midpoint of the interval means that the maximum error is less than half the interval itself. I illustrate this by taking a second look at the series from Example 5, and show that with just 10 terms of the partial sum, we obtain the same accuracy that we had previously done with 32 terms.

#math #calculus #series #approximation #education

Vector Functions: Problems Plus - Notes - YouTube - Playlist - Vector Functions - mes.fm/math

In this video, I go over the end-of-chapter Problems Plus questions and solve 7 very advanced problems on vector functions. Three of the questions involve launching a projectile either on a flat, inclined, or declined plane. One question is on curvature, one on finding the equation of a plane, another is on rolling a ball off a table, and my favorite one is on finding the axial distance covered when wrapping a cable of known length around a spool of known diameter. Buckle up for 5 hours of mathematics!

#math #calculus #vectorfunctions #problemsplus #education

Problems Plus 1: Projectile fired from the origin - Notes - YouTube - Playlist - Vector Functions - mes.fm/math

In this video, I examine projectile motion launched from the origin and show that the projectile will always lie within a parabola defined by launching a projectile from the maximum height horizontally at the same initial velocity. I also show that if we are to aim directly at a target downrange and drop the target at the same instance of firing the projectile, then we will always hit our target!

#math #calculus #vectorfunctions #problemsplus #projectilemotion

Projectile aimed at a target downrange will always hit when dropped and fired at the same time! - Notes - Problems Plus - mes.fm/math

In this video, I show that firing a projectile while aiming at a distance target downrange and dropping it at the same time will always cause the projectile to hit the target, provided that the projectile doesn't hit the ground before reaching the target. This is because both the projectile and object accelerate downwards at the same rate due to gravity, and this is represented in the (-1/2)gt² term of the y-component in the parametric equations of projectile motion. This result is pretty amazing, and allows for some pretty cool stunts and trick shots to attempt!

#math #calculus #vectorfunctions #problemsplus #projectilemotion

Problems Plus 2: Firing a Projectile Up and Down an Inclined Plane - Notes - Problems Plus - mes.fm/math

In this video, I go over projectile motion along an inclined plane and show that the launch angle that gives the maximum range upwards or downwards is half the angle between the plane and the vertical. Even when firing at a target up an inclined plane, the angle that minimizes the energy used is also half the angle between the plane and the vertical. This result matches for a flat plane, since the maximum distance is achieved for a launch angle of 45°, i.e. half the angle between the plane (0°) and the vertical (90°).

#math #calculus #projectilemotion #vectors #vectorfunctions

Problems Plus 3: Physics of a Ball Rolling Off a Table - Notes - Problems Plus - mes.fm/math

In this video, I go over the physics of a ball rolling off a table and determine how far it falls, the angle and speed it bounces with, and the distance it travels for a second bounce after losing 80% of its energy upon impact.

#math #calculus #projectilemotion #vectors #vectorfunctions

Problems Plus 4: Curvature from Parametric Equations that involve Integrals - Notes - Problems Plus - mes.fm/math

In this video, I show how to obtain the curvature of parametric equations that involve integrals, by first utilizing the fundamental theorem of calculus to obtain the derivative of the position vector from the integrands of the integrals. Then we can follow the usual steps to obtain the curvature, which is defined as the magnitude of the unit tangent vector divided by the magnitude of the derivative of the position vector.

#math #calculus #curvature #vectors #vectorfunctions

Problems Plus 5: Launch Angle of 56° maximizes TOTAL distance a projectile travels - Notes - Problems Plus - mes.fm/math

In this video, I show that firing a projectile has a maximum total distance traveled in the air when the launch angle is approximately 56°. This is 11° higher than the 45° angle needed to maximize the total horizontal distance. I derive this by starting from the parametric equations of trajectory, obtaining the velocity vector as the derivative of the position vector, and then obtaining the integral formula for the arc length the projectile travels (whose integrand is the magnitude of the velocity vector). The arc length is maximized when its derivative is zero, i.e. at a critical point, thus obtaining our answer of about 56° and a max total distance of about 1.20v²/g. Fascinating stuff!

#math #calculus #physics #vectors #projectilemotion

Problems Plus 6: Cable wound around a spool - Notes - Problems Plus - mes.fm/math

In this video, I calculate the height covered by a coil of cable of known length around a spool, counting only the portion of the height that includes a full loop.

#math #calculus #physics #vectors #electromagnetism