   Chapter 13.2, Problem 7E

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.7. r(t) = 4 sin t i − 2 cos  t j, t = 3π/4

(a)

To determine

To sketch: The plane curve with the vector equation r(t)=4sinti2costj,t=3π4 .

Explanation

Write the x-component of the vector r(t)=4sinti2costj .

x=4sint

Rearrange the expression as follows.

x4=sint

Take square on both sides.

(x4)2=sin2(t) (1)

Write the y-component of the vector r(t)=4sinti2costj .

y=2cost

Rearrange the expression y=2cost as follows.

y2=cost

Take square on both sides of the expression.

(y2)2=(cost)2

(y2)2=cos2(t) (2)

Add equation (1) with equation (2) as follows.

(x4)2+(y2)2=sin2(t)+cos2(t)x216+y24=1

The curve x216+y24=1 is an ellipse.

The required curve is a graph of x216+y24=1 .

Find the vector r(t) at t=3π4

(b)

To determine

To find: The vector r(t) from the vector function r(t)=4sinti2costj at t=3π4 .

(c)

To determine

To sketch: The position vector r(t) and the tangent vector r(t) at t=3π4 .

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