A point on an ellipse with major axis length 2a and minor axis length 2b has the coordinates \((a \cos \theta, b \sin \theta)\), \(0 \leq \theta \leq 2 \pi\).

(a) Show that the distance from this point to the focus at (-c, 0) is \(d(\theta)=a+c \cos \theta\), where \(c=\sqrt{a^{2}-b^{2}}\)

(b) Use these coordinates to show that the average distance \(\bar{d}\) from a point on the ellipse to the focus at (-c, 0), with respect to angle \(\theta\), is a.

Text Transcription:

(a cos theta, b sin theta)

0 leq theta leq 2 pi

d(theta)=a+c cos theta

c=sqrt a^2-b^2

bar d

theta

Module #18-‐ Basic concepts of sensation and perception Sunday, February 28, 2016 5:55 PM - Heather Sellersc 'urious mix of "perfect vision" and face blindness: the distinction between sensation and perception ○ Sensation: the process by which our sensory receptors and nervous system receive and represent stimulus energies from our environment (sensory receptors detect information→nervous system transmits the information to her brain) [normal] ○ Perception: the process of organizing and interpreting sensory information, enabling us to recognize meaningful objects and events [almost normal] ○ Bottom -‐up processing: analysis that begins with the sensory receptors and works up to the brain's integration of sensory information [build up from smallest pieces of sensory info/detect lines, angels, colors…] ○ Top-‐down processing: constructs perceptions from the sensory input by drawing on our experience and expectations [perception that's driven by perception/ the brain applies what y