   Chapter 10.2, Problem 18E

Chapter
Section
Textbook Problem

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work.18. x = t3 − 3t, y = t3 − 3t2

To determine

To find: The tangent is horizontal or vertical and plot the curve for the parametric equations x=t33t and y=t33t2 .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=t33t (1)

The parametric equation for the variable y is as follows.

y=t33t2 (2)

Calculation:

Differentiate the parametric equation for x with respect to t .

x=t33t

dxdt=3t23

Differentiate the parametric equation for y with respect to t .

y=t33t2dydt=3t26t

The tangent is horizontal when the expression dydt=0 is true.

dydt=03t26t=03t(t2)=0t=0andt=2

Substitute 0 for t in equation (1).

x=t33t=(0)33(0)=0

Substitute 0 for t in equation (2).

y=t33t2=(0)33(0)2=0

Substitute 1 for t in equation (1).

x=t33t=(2)33(2)=2

Substitute 2 for t in equation (2).

y=t33t2=(2)33(2)2=4

The curve has horizontal tangent touching at points (0,0) and (2,4)

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