Show That If $\mathbf { r } ^ { \prime } ( t )$

question 70

Essay

Show that if $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are parallel at some point on the curve described by $\mathbf { r } ( t )$ , then the curvature at that point is 0. Give an example of a curve $\mathbf { r } ( t )$ for which $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are always parallel.

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Definitions:

Visual Area

A region of the brain, specifically in the occipital lobe, responsible for processing visual information.

Optic Tract

A part of the visual system in the brain, consisting of a bundle of nerve fibers that transmits visual information from the optic chiasma to the thalamus, hypothalamus, and midbrain.

Visual Field

The entire area that can be seen when the eyes are fixed in one position, including both central and peripheral vision.

Sense of Smell

The ability to perceive odors, a chemical sensing system also known as olfaction.