   Chapter 10.3, Problem 41E

Chapter
Section
Textbook Problem

# Determining Concavity In Exercises 43-48, determine the open t-intervals on which the curve is concave downward or concave upward. x = 2 t + ln t , y = 2 t − ln t

To determine

To calculate: The open t-interval on which the curve of parametric equations, x=2t+lnt,y=2tlnt, is concave downward or concave upward.

Explanation

Given:

The parametric equation,

x=2t+lnty=2tlnt

Formula used:

If a smooth curve is given by the parametric equations x(t) and y(t), then the slope of the curve is,

dydx=dy/dtdx/dt, dxdt0

Calculation:

Consider the equations,

x=2t+lnty=2tlnt

Differentiate x=2t+lnt with respect to t, to get,

dxdt=2+1t ...... (1)

Differentiate y=2tlnt with respect to t, to get,

dydt=21t ...... (2)

Divide equation (2) by (1), to get,

dydx=dydtdxdt=21t2+1t=2t12t+1

Now, Differentiate again, to get,

d2ydx2=ddt(2t12t+1)

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