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Chapter 9: Q.7 (page 721)

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5鈥�8.

x=et,y=lnt,t>0

Short Answer

Expert verified

The curve rises and goes to the right.

Step by step solution

01

Step: 1 Given information

Consider the parametric equationsx=et,y=lnt,t>0

02

Calculation

Consider the parametric equations x=e',y=lnt,t>0

The goal is to examine the parametric equations' direction of motion.

Take the function x=e'.

Differentiate with respect to t.

Then, x3(t)=ddtet

x'(t)=e'>0

03

Further calculation

Take the function now.y=lnt.

different in terms oft.

y3(t)=ddt(lnt)y3(t)=1t>0

Thus, x'(t)>0and y'(t)>0.

The value of t, the derivatives of the functions x(t), y(t)are also growing. As a response, the curve goes up and to the right throughout its motion.

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