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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

At what point does the curve have maximum curvature? What happens to the curvature as x ⟶ ∞?

30. y = ln x

31. y = ex

To determine

the point of curve y=ex that possess maximum curvature and the curvature when x tends to infinity.

Explanation

Given data:

Vector function is y=ex .

Formula used:

Consider a vector function of the curve in the form of y=f(x) .

Write the expression for curvature.

k(x)=|f(x)|[1+(f(x))2]32 (1)

Here,

f(x) is function of x,

f(x) is first derivative of function f(x) , and

f(x) is second derivative of function f(x) .

The equation y=ex indicates y is a function of x. Hence consider y=f(x) .

Write the vector function of curve.

y=ex (2)

Substitute f(x) for y,

f(x)=ex

Apply differentiation with respect to x on both sides of equation.

f(x)=ddx(ex) {ddt(et)=et}=ex

Apply differentiation with respect to x on both sides of equation.

f(x)=ddx(ex) {ddt(et)=et}=ex

Substitute ex for f(x) and ex for f(x) in equation (1),

k(x)=|ex|[1+(ex)2]32=ex(1+e2x)32

k(x)=ex(1+e2x)32 (3)

The point of maximum curvature of the curve occurs at value of x, when k(x)=0 .

Apply differentiation with respect to x on both sides of equation (3).

k(x)=ddx(ex(1+e2x)32)=(ddxex)(1+e2x)32+ex(ddx(1+e2x)32){ddt(uv)=uv+uv}=ex(1+e2x)32+ex(32(1+e2x)321(2e2x)){

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