Math Problem Statement

Consider the curve described by the equation x p/q + y p/q = 1. (a) Find the point(s) where the slope of the tangent line is equal to 1 if p q = 4/3

Solution

To analyze the curve described by the equation

xpq+ypq=1x^{\frac{p}{q}} + y^{\frac{p}{q}} = 1

with the given condition that pq=43\frac{p}{q} = \frac{4}{3}, let’s follow these steps:

  1. Substitute pq=43\frac{p}{q} = \frac{4}{3} into the equation:

    x43+y43=1x^{\frac{4}{3}} + y^{\frac{4}{3}} = 1

  2. Implicit Differentiation: To find points where the slope of the tangent line is equal to 1, we need to find dydx\frac{dy}{dx} by differentiating both sides of the equation with respect to xx.

    Differentiating x43+y43=1x^{\frac{4}{3}} + y^{\frac{4}{3}} = 1 with respect to xx:

    43x431+43y431dydx=0\frac{4}{3}x^{\frac{4}{3} - 1} + \frac{4}{3}y^{\frac{4}{3} - 1} \cdot \frac{dy}{dx} = 0

    Simplify this to:

    43x13+43y13dydx=0\frac{4}{3}x^{\frac{1}{3}} + \frac{4}{3}y^{\frac{1}{3}} \cdot \frac{dy}{dx} = 0

  3. Solve for dydx\frac{dy}{dx}:

    Rearranging terms,

    dydx=x13y13\frac{dy}{dx} = -\frac{x^{\frac{1}{3}}}{y^{\frac{1}{3}}}

  4. Set Slope Equal to 1: Now, we are asked to find the points where the slope of the tangent line is equal to 1. Therefore, set dydx=1\frac{dy}{dx} = 1:

    x13y13=1-\frac{x^{\frac{1}{3}}}{y^{\frac{1}{3}}} = 1

    Solving this equation:

    x13=y13x^{\frac{1}{3}} = -y^{\frac{1}{3}}

    Since x43+y43=1x^{\frac{4}{3}} + y^{\frac{4}{3}} = 1 only has solutions for real values when xx and yy are positive, no real solution exists for points where the slope is exactly 1 in this scenario.

Let me know if you'd like more details on each step or have any questions. Here are some related questions to explore this topic further:

  1. How does implicit differentiation work with more complex exponents?
  2. What happens if pq\frac{p}{q} has a different rational value?
  3. How can we interpret solutions in the complex plane?
  4. What other curves have similar symmetrical properties?
  5. How do we find normal lines instead of tangents for such curves?

Tip: When dealing with equations involving rational exponents, be cautious of sign constraints that limit real solutions.

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Curve Analysis
Exponent Rules

Formulas

x^(p/q) + y^(p/q) = 1
Implicit differentiation to find dy/dx

Theorems

Implicit Differentiation

Suitable Grade Level

Undergraduate