   Chapter 10.2, Problem 10E

Chapter
Section
Textbook Problem

Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent.10. x = sin πt, y = t2 + t; (0, 2)

To determine

To find: The equation of the tangent for the parametric equations x=sinπt and y=t2+t , plotting the graphs for the curve and tangent.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=sinπt (1)

The parametric equation for the variable y is as follows.

y=t2+t (2)

Calculation:

Differentiate the parametric equation x with respect to t .

x=sinπtdxdt=πcosπt

Differentiate the parametric equation y with respect to t .

y=t2+tdydt=2t+1

Write the chain rule for dydx .

dydx=dydtdxdt

Substitute (2t+1) for dydt and (πcosπt) for dxdt in the above equation.

dydx=(2t+1)(πcosπt) (3)

To find t , solve the expression y=2 .

y=2t2+t2=0(t+2)(t1)=0t=2ort=1

Substitute (1) for t in equation (3)

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