r(t) = < e^t * cost, e^t*sint, 5>
Find the curvature at the point (x, y, z) = (-e^π, 0, 5)
explain briefly how you come up with answer and also express answer as a decimal.
If a point belongs to a curve then exists a parameter value such that
. In case of a given point
Wikipedia says that for a space curve its curvature is given by
.
All derivatives are with respect to variable t. Here is a function in variable , Find the expression for and evaluate it for .
Also, notice that as z is constant you are really looking at,
... so you only need the 2-D version (Curvature - Wikipedia, the free encyclopedia).
Just in case a picture helps differentiate...
... where (key in spoiler) ...
Spoiler:
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!