By: Himali Rajakarunanayake, Jasmine Kriston, Jessica HoriPeriod 5

An object or drawing which appears to have properties which are physically impossible, deceptive, or counterintuitive.

Calculus definition: an optical illusion is created by having lines intersect a family of curves. In each case the lines appear to be curved.

What is a Slope Field? – A graphical representation of the solutions of a differential equation.

-A differential equation determines the slope of the solution at any point.

Family of Curves: formed when slope field isshown. When constant (C) is given, the particular equation is shown.

tch/10215

(a) x2 + y2 = C2 ; x=3, y=4, C=5

x2 + y2 = 25 Implicit Differentiation: 2x + 2y(dy/dx) =0 dy/dx=(-2x/2y) Plug in x and y values: dy/dx= (-3/4)

(b) xy=C ; x=1, y=4, C=4 xy=4 Implicit Differentiation: y + x(dy/dx) = 0 dy/dx= (-y/x) Plug in x and y values: dy/dx= -4

The St. Louis Gateway Arch is an optical illusion because it appears to be taller than it is wide. However…

However, the distances of the height and the width are both 630 feet. It is explained by this mathematical equation:

y= 693.8507-68.7672 (( e 0.0100333x + e -.0100333x )) (2) dy/dx= -0.3449809739(e0.0100333x + e-.0100333x)

Initial condition f(0)= 625.0835

http://www.sciencefriday.com/videos/watch/10215

Calculus of a Single Variable 8th

Edition by Larson, Hosteller, and Edwards (textbook)

CRC concise encyclopedia of mathematics by Eric Weisstein

Queen Mary University Of London Website: http://www.agenarisk.com/resources/probability_puzzles/St%20Louis%20Arch.shtml

http://www.sciencefriday.com/videos/watch/10215