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
with the given condition that , let’s follow these steps:
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Substitute into the equation:
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Implicit Differentiation: To find points where the slope of the tangent line is equal to 1, we need to find by differentiating both sides of the equation with respect to .
Differentiating with respect to :
Simplify this to:
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Solve for :
Rearranging terms,
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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 :
Solving this equation:
Since only has solutions for real values when and 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:
- How does implicit differentiation work with more complex exponents?
- What happens if has a different rational value?
- How can we interpret solutions in the complex plane?
- What other curves have similar symmetrical properties?
- 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
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