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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter.

8. x = 1 + t , y = e t 2 ; (2, e)

(a)

To determine

To find: The equation of the tangent without eliminating the parameter for the parametric equations x=1+t and y=et2 .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=1+t (1)

The parametric equation for the variable y is as follows.

y=et2 (2)

Calculation:

Differentiate the parametric equation (1) for x with respect to t .

x=1+tdxdt=12t

Differentiate the parametric equation (2) for y with respect to t .

y=et2dydt=et22t

Write the chain rule for dydx .

dydx=dydtdxdt

Substitute (et22t) for dydt and (12t) for dxdt in the above equation.

dydx=(et22t)[12t]=4t3/2et2 (3)

Write the equation for tangent

(b)

To determine

To find: The equation of the tangent by first eliminating the parameter for the parametric equations x=1+t and y=et2 .

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