   Chapter 13.2, Problem 4E

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.4. r(t) = ⟨t2, t3⟩, t = 1

(a)

To determine

To sketch: The plane curve with the vector equation r(t)=t2,t3,t=1.

Explanation

Write the x-component of the vector r(t)=t2,t3.

x=t2

Rearrange the expression x=t2 as follows.

t=x

Write the y-component of the vector r(t)=t2,t3.

y=t3 (1)

Substitute x for t in equation (1),

y=(x)3=(x12)3=x32

Modify the expression as follows.

x=y23

The required curve is a graph of x=y23.

Find the vector r(t) at t=1

(b)

To determine

To find: The vector r(t) from the vector function r(t)=t2,t3,t=1.

(c)

To determine

To sketch: The position vector r(t) and the tangent vector r(t) at t=1.

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