Calculus Animations,Graphics and Lecture Notes

Optimization and Related Rates

About Kelly Liakos
Calculus 1- Limits and Derivatives
Calculus 1 - The Second Derivative
Calculus 1 and 3 Formula Sheets
Chain Rule
Computer Lab Assignments
Conic Sections
Differential Equations
Directional Derivatives/Gradient
Double Integrals
Equillibrium Solutions 1st order DEs
1st order Diff Eqs -Motion
Flux Integrals and Surface Integrals
Infinite Sequences
Infinie Series
Level Curves and Level Surfaces
Line Integrals
Optimization and Related Rates
Optimization for Functions of 2 Variables
Parametric Equations 2-space
Parametric Equations 3-space
Partial Derivatives
Polar Coordinate System
Polar Coordinates- Derivatives and Integrals
Riemann Sums and the Fundamental Theorem of Calculus
2d order Diff EQS-Motion
2d Partial Derivatives
Supplemental Exercises and Solutions
Tangent Planes/ Differential for f(x,y)
Trigonometry Applications
Triple Integrals
Unit Tangent Vectors/Unit Normal Vectors
Vectors in General
Vector Valued Functions
Visualizing Limits of Functions of 2 Variables
Special Topics
Scratch Paper
2 Poems

In the coming days I'll be adding many more examples.

Related Rates

Suppose there is a light at the top of a pole 50 ft high. An object is 30 feet from the base of the light

pole and is released from a height of also 50 ft. Find the rate at which the shadow is moving along the


See Animation Moving Shadow and the Moving Shadow Notes for the solution

Animation - Moving Shadow

Notes- solution to the moving shadow problem

The following notes and animations deal with a ship and a submarine and blowing up the submarine.

Notes - The Ship and the Submarine

Animation 1

Animation 2

The following notes and animations deal with  shortening and lengthening of shadows as a person approaches or walks away from a lightpole. You may want to view the animations before the solution to problems involving thios situation.

Animation-Walking toward the light

Animation-Walking awayfrom the light

Notes- Solution to Walking Towards the Light

A falling light source -one for you to do.

The following animation refers to the following problem which appeared on the Mathhelp Forum

(an excellent  Calculus resource):

A light source is dropped from a height of \frac{1}{2}gT^2  at  t=0.
From a point on the ground which is 
distance a from the tower base, a stone is thrown with velocity of
gT vertically at the same time.
Show the velocity of the stone's shadow on the ground is 13a/(2T)  when t=T/4

In the animation I use T =6m and a = 100m

Animation - A Falling Light Source

The following are the solutions and animation which deals with the area between expanding concentric circles and the rate of change of thickness of a spherical bubble. Again you may want to view the animation first.

Notes - Expanding Circles and Spherical Bubbles

Animation-Expanding Circles

Notes - A Ferris Wheel Problem

Animation- Ferris Wheel

Related Rates and Running Down the Beach

Animation-Running down the Beach

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

Animation-3rd Base

Animation-2d Base


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 motion.

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

Animation-No Damping

Animation-Damped Motion

Minimization of Area


Let y1(x) = a3x2 - a4x and y2(x) = x

a. 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

Animation Triangle

Solution to Triangle Problem

The Smart Bunny-A very short story by Kurt Vonnegut Jr.

©2008-2010 Kelly Liakos

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