   Chapter 13, Problem 12RE

Chapter
Section
Textbook Problem

# Find the curvature of the ellipse x = 3 cos t, y = 4 sin t at the points (3, 0) and (0, 4).

To determine

To find: The curvature of the ellipse x=3cost,y=4sint at the points (3,0) and (0,4) .

Explanation

Formula used:

Write the expression to find the curvature k(t) with the vector function r(t) .

k(t)=|r(t)×r(t)||r(t)|3 (1)

Here,

r(t) is the tangent vector, which is the derivative of vector r(t) .

Write the required differential formulae to find the tangent vector r(t) as follows.

ddtsint=costddtcost=sint

Write the vector function r(t) from the equations of the ellipse x=3cost,y=4sint as follows.

r(t)=3cost,4sint

Calculation of tangent vector r(t) :

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=3cost,4sint as follows.

ddt[r(t)]=ddt3cost,4sintr(t)=ddt(3cost),ddt(4sint)

Rewrite and compute the expression as follows.

r(t)=3ddt(cost),4ddt(sint)=3(sint),4cost=3sint,4cost

Calculation of vector r(t) :

Differentiate each component of the vector function r(t)=3sint,4cost as follows.

ddt[r(t)]=ddt3sint,4costr(t)=ddt(3sint),ddt(4cost)

Rewrite and compute the expression as follows.

r(t)=3ddt(sint),4ddt(cost)=3cost,4(sint)=3cost,4sint

Calculation of vector r(t)×r(t) :

Substitute 3sint,4cost for r(t) and 3cost,4sint for r(t) in the expression r(t)×r(t) ,

r(t)×r(t)=3sint,4cost×3cost,4sint

Rewrite the expression as follows.

r(t)×r(t)=[(3sint)i+(4cost)j+(0)k]×[(3cost)i+(4sint)j+(0)k]

Rewrite and compute the expression as follows.

r(t)×r(t)=|ijk3sint4cost03cost4sint0|={i[(4cost)(0)(4sint)(0)]j[(3sint)(0)(3cost)(0)]+k[(3sint)(4sint)(3cost)(4cost)]}=(00)i(00)j+(12sin2t+12cos2t)k=(12)(sin2t+cos2t)k

Simplify the expression as follows

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