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(dy = arctan dx, (i) Use the facts: dy and (dy/dt) to find dt d dx (dx/dt) ds (ii) Use the fact that s(t) = *² + ÿ² dt to find dt to dø (аф/dt) (iii) Apply the Chain Rule to conclude that ds (ds/dt) (b) By regarding a curve y=f (x) as the parametric curve: x= x , y=f(x ), with parameter x, show that the formula from part (a) reduces to |d²y/dx² | K = [1 + (dy/dx)² ]3/2 *

(dy = arctan dx, (i) Use the facts: dy and (dy/dt) to find dt d dx (dx/dt) ds (ii) Use the fact that s(t) = *² + ÿ² dt to find dt to dø (аф/dt) (iii) Apply the Chain Rule to conclude that ds (ds/dt) (b) By regarding a curve y=f (x) as the parametric curve: x= x , y=f(x ), with parameter x, show that the formula from part (a) reduces to |d²y/dx² | K = [1 + (dy/dx)² ]3/2 *

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 20T
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The problem is rather long so I have attached pictures of the instructions below.

dø The curvature at a point P of a curve is defined as K = ds where o is the angle of inclination of the tangent line at P, as shown in the figure. Thus, the curvature is the absolute value of the rate of change of ¢ with respect to arc length. It can also be regarded as a measure of the rate of change of the direction of the yA curve at P. (a) For the parametric curve x = x ( t), y=y ( t ), derive the formula |à ÿ – * ỷ| [x2 + y?]3/2 K = dx where the dots indicate the order of the derivative with respect to t, so = dt To accomplish this task, carry out the following steps:
Transcribed Image Text:dø The curvature at a point P of a curve is defined as K = ds where o is the angle of inclination of the tangent line at P, as shown in the figure. Thus, the curvature is the absolute value of the rate of change of ¢ with respect to arc length. It can also be regarded as a measure of the rate of change of the direction of the yA curve at P. (a) For the parametric curve x = x ( t), y=y ( t ), derive the formula |à ÿ – * ỷ| [x2 + y?]3/2 K = dx where the dots indicate the order of the derivative with respect to t, so = dt To accomplish this task, carry out the following steps:
dø to find dt (dy/dt) (i) Use the facts: ´dy` = arctan \dx, dy and dx (dx/dt) ds (ii) Use the fact that s(t) = | *2 + y2 dt to find dt dø (dф/dt) (iii) Apply the Chain Rule to conclude that ds (ds/dt) (b) By regarding a curve y= f(x) as the parametric curve: x= x , y=f(x ), with parameter x , show that the formula from part (a) reduces to |d?y/dx? | K = [1 + (dy/dx)² ]3/2 *
Transcribed Image Text:dø to find dt (dy/dt) (i) Use the facts: ´dy` = arctan \dx, dy and dx (dx/dt) ds (ii) Use the fact that s(t) = | *2 + y2 dt to find dt dø (dф/dt) (iii) Apply the Chain Rule to conclude that ds (ds/dt) (b) By regarding a curve y= f(x) as the parametric curve: x= x , y=f(x ), with parameter x , show that the formula from part (a) reduces to |d?y/dx? | K = [1 + (dy/dx)² ]3/2 *
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