Lab 4 -- Math 4C: Curvature NAME
______________________
Find the circle of best fit for the curve y = arctan (x) at the point
P(1,p/4); that is, the circle that is tangent
to the curve at that point and has the same curvature.
Steps:
1. Find y¢ and
y¢¢ and evaluate them both at the point P.
y'
______________________ y' at P
________________________
y''
______________________ y'' at P
_______________________
2. Find the curvature k at P __________________
3. Find the radius of curvature ρ at
P ___________________
4. Interpret y¢ as
slope Δy / Δx, and use the result to find a tangent vector at
P: <Δx,Δy>
Tangent vector at P:
________________________
5. Find the slope of the normal line, and
use it to find a normal vector pointing toward the center of the circle.
Normal vector at P:
__________________________
6. Find the unit
normal vector N at P. _________________
7. Find the center (h,k) of
the circle: From P, move a distance ρ
along N.
Center (<h,k> = vector
OP + ρN): ____________________
8. Write parametric
equations of the circle (See lesson 1.7, figure 10 and formula 6):
x = _______________, y
= ________________, z = _________________
9. Use Maple to graph both
the arctangent curve and the circle on the same graph, using the “constrained”
viewing option. Print the
graph and turn it in with this paper.
Look at your graph to confirm that your circle in fact passes through
the point P and has the same tangent line and the same curvature.
10. Differentiate the
equation of your circle implicitly and confirm algebraically that it has the
same y¢ as the curve at the point P.
11. Find the second derivative and
confirm algebraically that the circle has the same y¢¢ as the arctangent curve at the point P
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Updated 07/10/06 by Don Power