   Chapter 13.3, Problem 45E

Chapter
Section
Textbook Problem

# Use the formula in Exercise 42 to find the curvature.45. x = et cos t, y = et sin t

To determine

To find: The curvature of the plane with parametric equations x=etcost and y=etsint .

Explanation

Given data:

Parametric equations are x=etcost and y=etsint .

Formula used:

Consider the expression of curvature of plane with parametric equation x=f(t) , y=g(t) .

k(t)=|x˙y¨x¨y˙|(x˙2+y˙2)32 (1)

Here,

k(t) is curvature of the plane.

Consider the given equation.

x=etcost

Differentiate the equation with respect to t.

x˙=ddt(etcost)=(ddtet)cost+et(ddtcost) {ddx(uv)=uv+uv}=etcost+et(sint) {ddxex=ex,ddx(cosx)=sinx}=et(costsint)

Find the value of x¨ .

x¨=ddt(et(costsint))=(ddt(et))(costsint)+et(ddt(costsint)) {ddx(uv)=uv+uv}=et(costsint)+et(sintcost) {ddxex=ex,ddx(cosx)=sinx,ddx(sinx)=cosx}=2etsint

Consider the given expression.

y=etsint

Differentiate the equation with respect to t.

y˙=ddt(etsint)=(ddtet)sint+et(ddtsint) {ddx(uv)=uv+uv}=etsint+et(cost) {ddxex=ex,ddx(sinx)=cosx}=et(sint+cost)

Find the value of y¨

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