     Calculus Animations,Graphics and Lecture Notes Optimization and Related Rates    Home About Kelly Liakos Sponsors 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 PreCalculus 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 Trigonometry Applications Triple Integrals Unit Tangent Vectors/Unit Normal Vectors Vectors in General Vector Valued Functions Visualizing Limits of Functions of 2 Variables Work Links 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

ground.

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

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 at  t=0.
From a point on the ground which is
distance a from the tower base, a stone is thrown with velocity of 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

Optimization

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) = xa. Find the value of a minimizes the areab. 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.  