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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?

11. x = t2 + 1, y = t2 + t

To determine

To find: The expression of dydx and d2ydx2 for the parametric equations x=t2+1 and y=t2+t , and find the value of t for which the curve is concave upward.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=t2+1 (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=t2+1dxdt=2t

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 (2t) for dxdt in the above equation.

dydx=(2t+1)(2t)=1+12t

Calculate the second derivative to determine concavity.

d2ydx2=ddt(dydx)dxdt=(1(2t2))2t=14t3

Substitute (1) for t in equation (1)

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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