   Chapter 10.3, Problem 39E

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 = 3 t 2 , y = t 3 − t

To determine

To calculate: The interval of t on which the curve of parametric equations, x=3t2,y=t3t is concave downward or concave upward.

Explanation

Given:

Consider the parametric equation,

x=3t2y=t3t

Formula used:

Differentiation of some standard functions are

d(xn)dx=nxn1, d(cf(x))dθ=cd(f(x))dθ

Calculation:

Consider the equations,

x=3t2y=t3t

Differentiate the expression x=3t2 with respect to t, to obtain,

dxdt=6t ...... (1)

Differentiate the expression y=t3t with respect to t, to obtain,

dydt=3t21 ...... (2)

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

dydx=dydtdxdt=3t216t=t2

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