The problem is rather long so I have attached pictures of the instructions below. døThe curvature at a point P of a curve is defined as K =dswhere 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 respectto arc length. It can also beregarded as a measure of the rateof change of the direction of theyAcurve at P.(a) For the parametric curvex = x ( t), y=y ( t ),derive the formula|à ÿ – * ỷ|[x2 + y?]3/2K =dxwhere the dots indicate the order of the derivative with respect to t, so =dtTo accomplish this task, carry out the following steps: døto finddt(dy/dt)(i) Use the facts:´dy`= arctan\dx,dyanddx(dx/dt)ds(ii) Use the fact that s(t) = |*2 + y2 dt to finddtdø(dф/dt)(iii) Apply the Chain Rule to conclude thatds(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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Description: The problem is rather long so I have attached pictures of the instructions below. døThe curvature at a point P of a curve is defined as K =dswhere o is the angle of inclination of the tangent line at P, as shown in the figure.Thus, the curvature is t

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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

Calculus: Early Transcendentals

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ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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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:

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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