   Chapter 13.2, Problem 2E

Chapter
Section
Textbook Problem

# (a) Make a large sketch of the curve described by the vector function r(t) = ⟨t2, t), 0 ≤ t ≤ 2, and draw the vectors r(1), r(1.1), and r(1.1) − r(1).(b) Draw the vector r'(1) starting at (1, 1), and compare it with the vector r ( 1.1 )   -    r (1) 0.1 Explain why these vectors are so close to each other in length and direction.

(a)

To determine

To sketch: The curve described by the vector function r(t)=t2,t,0t2 along with the vectors r(1), r(1.1), and r(1.1)r(1).

Explanation

From the vector function r(t)=t2,t,0t2, the x-component of the vector is t2 and the y-component of the vector is t.

The parametric equations from the vector function r(t)=t2,t,0t2 are written as follows.

x=t2,y=t,0t2

Substitute y for t in the expression x=t2.

x=y2

The curve of the vector function r(t)=t2,t,0t2 is equivalent to the curve x=y2.

The point at which the vector r(t) intersects the curve r(t)=t2,t,0t2 is (t2,t).

The point at which the vector r(1) intersects the curve r(t)=t2,t,0t2 is determined as follows.

Substitute 1 for t in the point (t2,t).

(t2,t)=(12,1)=(1,1)

The vector r(1) is joined from the origin to the point (1,1)

(b)

To determine

To draw: The vector r(1) such that it starts from the point (1.1) and compare the vector r(1) with the vector r(1.1)r(1)0.1.

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