The following are my notes and animations on related
rates problems involving baseball.
Note we can also use this format
for related rate problems involving planes, boats, and cars traveling on different bearings
Notes - Related Rates- A Baseball Example
The following are my notes and animation on the optimization
problem -a bird flies home.
Notes - A Bird Flies Home
Animation-A bird flies home
The following notes and animations deal with projectile
The notes solve the problem of what firing angle maximizes
the range of a projectile fired from ground level.
We first solve
the problem if the projectile is fired on flat ground. The seond deals with firing the projectile up a ramp.
The animations correspond to these 2 situations.
Notes - Maximizing the Range of a Projectile
Animation - Projectile Fired on Level Ground
Animation - Projectile Fired up a Ramp
The following notes and animation deal with minimizing the
distance from a point to an ellipse.
Notes-Minimizing the distance from a point to an ellipse
Animation-Minimum distance from a point to an ellipse
The following notes and animations use the product rule to find the maximum and minimum
heights of a mass on a spring where air resistance decreases the amplitude--this is known as the damped harmonic oscillator.
Max and Min Values of a Damped Mass on a Spring
Minimization of Area
Let y1(x) = a3x2 - a4x and y2(x) = x
Find the value of a minimizes the area
b. What is the minimum area?
Solution to Area Minimization Problem
Find the minimum area of the triangle formed by the tangent line to
f(x) = x2 +2 at (a,f(a)),
the x axis and the line segment from the point (a,f(a))
to the point (4,0)
For 0 < a < 4
See the Animation Triangle
Solution to Triangle Problem