   Chapter 10.3, Problem 24E

Chapter
Section
Textbook Problem

# Finding Equations of Tangent Lines In Exercises 27-30, find the equations of the tangent lines at the point where the curve crosses itself. x = 2 − π cos t ,   y = 2 t − π sin t

To determine

To calculate: The Equation of the tangent line at the point where curve crosses itself of the equation, x=2πcost and y=2tπsint.

Explanation

Given:

The parametric equation, x=2πcost and y=2tπsint.

Formula used:

If a smooth curve C given by the equation x=f(t) and y=g(t). Then the slope of the curve C is, dydx=(dy/dt)(dx/dt).

Calculation:

Consider the provided parametric function,

x=2πcost and y=2tπsint.

Procedure to draw the graph of the parametric equation through Ti-83,

Step1: Press ON key.

Step2: Press MODE key. Then select “Par” and press Enter key.

Step3: Press Y= key.

Step4: Enter the parametric equation “ X1=2πcost and Y1=2tπsint.

Step4: Set window key Tmin=2,Tmax=2,Tstep=0.1,Xmin=2,Xmax=5,Xscl=1,Ymin=2,Ymax=2 and Yscl=1.

Step5: Press TRACE key.

The graph of the parametric function is,

From the above figure, the curve crosses itself at the point (2,0). Now calculate the value of parameter (t) at that point.

Consider the equation,

x=2πcost

Substitute t=2 in the above equation and find the value of t,

2=2πcost0=πcostcost=0t=±π2

Again, consider the parametric function,

x=2πcost

Differentiate the above equation with respect to t,

dxdt=ddt(2πcost)=0π(sint)=πsint

Now, consider the parametric function,

y=2tπsint

Differentiate the above function with respect to t,

dydt=ddt(2tπsin

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