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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.

7. x = 1 + ln t, y = t2 + 2; (1,3)

(a)

To determine

To find: The equation of the tangent without eliminating the parameter for the parametric equations x=1+ln(t) and y=t2+2 .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=1+ln(t) (1)

The parametric equation for the variable y is as follows.

y=t2+2 (2)

Calculation:

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

dxdt=1+ln(t)=1t

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

dydt=t2+2=2t

Write the chain rule for dydx .

dydx=dydtdxdt

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

dydx=2t(1t)

dydx=2t2 (3)

Substitute 3 for y in equation (2)

(b)

To determine

To find: The equation of the tangent by first eliminating the parameter for the parametric equations x=1+ln(t) and y=t2+2 .

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