Math Problem Statement

Find \( \frac{dy}{dx} \) and \( \frac{d^2y}{dx^2} \) for the given parametric equations \( x = e^t \) and \( y = te^{-t} \). Determine for which values of \( t \) the curve is concave upward.

Solution

Let's solve the problem step-by-step.

Problem Summary:

We are given:

  1. x=etx = e^t
  2. y=te−ty = te^{-t}

We are tasked to:

  1. Find dydx\frac{dy}{dx}
  2. Find d2ydx2\frac{d^2y}{dx^2}
  3. Determine for which values of tt the curve is concave upward.

Solution

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Math Problem Analysis

Mathematical Concepts

Parametric Differentiation
Concavity
Calculus

Formulas

\( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \)
\( \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right) \)
Concavity determination using the second derivative

Theorems

Parametric Derivatives
Concavity Test

Suitable Grade Level

Grade 12 or College Calculus

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