Math Problem Statement

What are the properties of the function y = x / (x - 1)^2?

Solution

The equation shown in the image is:

y=x(x1)2y = \frac{x}{(x-1)^2}

Would you like me to differentiate it, analyze its behavior, find asymptotes, or anything else? Let me know how you would like to proceed.

Here are 5 related questions for further exploration:

  1. What is the derivative of yy with respect to xx?
  2. What are the critical points of yy?
  3. Can we determine the vertical and horizontal asymptotes of yy?
  4. What is the behavior of yy as xx \to \infty or x1x \to 1?
  5. Can yy have a maximum or minimum value?

Tip: Understanding the behavior of rational functions often requires analyzing their derivatives and asymptotes!

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Limits and Asymptotes
Critical Points
Derivatives

Formulas

Derivative of a quotient: d/dx [u/v] = (v * u' - u * v') / v^2
Limits for asymptotic behavior
y = x / (x - 1)^2

Theorems

Fundamental Theorem of Calculus
Limit Theorems for Rational Functions
Asymptote Analysis

Suitable Grade Level

Grades 11-12