OPTICAL ILLUSIONS:

WITH A CALCULUS

APPROACHMeghana Attaluri

Nikita Chaudhri

Karen Lo

Jessica Siu

3◦

WHAT’S THE CONCEPT? Tangent or secant lines to a family of

curves To find where the bending illusion

occurs, you must find the derivative of the line

The “bend” occurs at the tangent point and where lines intersect a family of lines/curves

BOOK PROBLEMS Circle: x2 + y2 = c2

x=3, y=4, c=5 x2 + y2 = 25 y2= 25-x2 y= (25-x2)1/2

dy/dx = ½ (25-x2)1/2(-2x) dy/dx = -x(25-x2)1/2

= -3(25-9)1/2 = -3/4

BOOK PROBLEMS Hyperbolas: xy=c x=1,y=4,c=4 xy=4 y=4x-1

dy/dx = -4/x2

dy/dx = -4/1 = -4

BOOK PROBLEMS Lines: ax=by x=31/2, y=3, a=31/2, b=1 y=ax/b dy/dx = a/b = 31/2

BOOK PROBLEMS Cosine curves: y=cosx x=/3, y=1/3, c=2/3 y= (2/3)cosx dy/dx = (-2/3)sinx = (-2/3)sin(/3) = (-2/3)(31/2/2) = -31/2/2

COOL CIRCLE ILLUSION

REAL WORLD APPLICATION Aviation school

Fashion

Architecture, interior design

Landscaping

psychology