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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 dy/dx and d2y/dx2. For which values of t is the curve concave upward?

12. x = t3 + 1, y = t2t

To determine

To find: The value of dydx and d2ydx2 for the parametric equations x=t3+1 and y=t2t .and find for value of t , whether the curve is concave upward.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=t3+1 (1)

The parametric equation for the variable y is as follows.

y=t2t (2)

Calculation:

Differentiate the parametric equation x with respect to t .

x=t3+1dxdt=3t2

Differentiate the parametric equation y with respect to t .

y=t2tdydt=2t1

Write the chain rule for dydx .

dydx=dydtdxdt

Substitute (2t1) for dydt and (3t2) for dxdt in the above equation.

dydx=(2t1)(3t2)=23t13t2

Calculate the second derivative to determine concavity.

d2ydx2=ddt(dydx)dxdt=(23t2+23t2)3t2=2(1t)3t33t2=2(1t)9t5

Substitute (1) for t in equation (1)

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