   Chapter 10.2, Problem 23E

Chapter
Section
Textbook Problem

Graph the curve in a viewing rectangle that displays all the important aspects of the curve.23. x = t4 − 2t3 − 2t2, y = t3 − t

To determine

To plot: The curve inside a viewing rectangle for important aspects of the curve for parametric equations x=t42t32t2 and y=t3t .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=t42t32t2 (1)

The parametric equation for the variable y is as follows.

y=t3t (2)

Calculation:

Differentiate the parametric equation x with respect to t .

dxdt=t42t32t2dxdt=4t6t24t (3)

Differentiate the parametric equation y with respect to t .

dydt=t3t

dydt=3t2t (4)

dydx=dydtdxdt

dydx=3t214t36t24t

Substitute 1 for t in equation (1).

x=t42t32t2=(1)42(1)32(t)2x=122x=3

Substitute 1 for t in equation (2).

y=t3t=(1)3(1)y=0

The values of x and y for each step value of t is tabulated in the below table.

 t 0 1 2 3 x 0 −3 −8 9 y 0 0 −1 −1

Graph:

The graph is plotted for the parametric equation in the viewing rectangle (2,1.1) by (0.5,0.5) is shown below in figure 1.

Use the condition for horizontal tangent line as below.

dydt=03t21=0t=±13

Substitute 13 for t in equation (1).

x=t42t32t2=(13)42(13)32(13)2x=0.170

Substitute 13 for t in equation (1).

x=t42t32t2=(13)42(13)32(13)2=0.940

Substitute 13 for t in equation (2).

y=t3t=(13)3(13)=+0

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