   Chapter 13.2, Problem 5E

Chapter
Section
Textbook Problem

# (a) Sketch the plane curve with the given vector equation.(b) Find r'(t).(c) Sketch the position vector r(t) and the tangent vector r'(t) for the given value of t.5. r(t) = e2t i + et j, t = 0

(a)

To determine

To sketch: The plane curve with the vector equation r(t)=e2ti+etj,t=0.

Explanation

Write the x-component of the vector r(t)=e2ti+etj,t=0.

x=e2t

Take natural log on both sides of the expression.

ln(x)=ln(e2t)

Rearrange the expression ln(x)=ln(e2t) as follows.

ln(x)=2tt=ln(x)2

Write the y-component of the vector r(t)=e2ti+etj,t=0.

y=et (1)

Substitute ln(x)2 for t in equation (1),

y=eln(x)2

Rewrite the expression as follows.

y=e0.5ln(x)=eln(x0.5)=x0.5

y=x12

Take square on both sides of the expression.

y2=(x12)2y2=xx=y2

The expression x=y2 is a part of parabola

(b)

To determine

To find: The vector r(t) from the vector function r(t)=e2ti+etj,t=0.

(c)

To determine

To sketch: The position vector r(t) and the tangent vector r(t) at t=0.

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