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Discriminant Calculator Calc 3

Discriminant Formula:

\[ D = f_{xx} \cdot f_{yy} - (f_{xy})^2 \]

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1. What is the Discriminant in Calculus 3?

The discriminant (D) is used in multivariable calculus to classify critical points of functions of two variables. It is calculated from the second partial derivatives of the function and helps determine whether a critical point is a local maximum, local minimum, or saddle point.

2. How Does the Calculator Work?

The calculator uses the discriminant formula:

\[ D = f_{xx} \cdot f_{yy} - (f_{xy})^2 \]

Where:

Interpretation:

3. Importance of the Discriminant

Details: The discriminant is crucial for analyzing the behavior of multivariable functions at critical points, which has applications in optimization problems across physics, engineering, and economics.

4. Using the Calculator

Tips: Enter the values of the second partial derivatives \( f_{xx} \), \( f_{yy} \), and the mixed partial derivative \( f_{xy} \). The calculator will compute the discriminant and you can interpret the result using the rules above.

5. Frequently Asked Questions (FAQ)

Q1: What if the discriminant equals zero?
A: When D = 0, the second derivative test is inconclusive. You may need to use other methods to analyze the critical point.

Q2: Can this be used for functions with more than two variables?
A: No, this discriminant is specifically for functions of two variables. Higher dimensions require more complex analysis.

Q3: How do I find the partial derivatives?
A: Partial derivatives are calculated by differentiating the function with respect to one variable while treating others as constants.

Q4: What's the difference between fxy and fyx?
A: For continuous functions with continuous partial derivatives (as typically encountered), fxy = fyx by Clairaut's theorem.

Q5: Can this calculator handle complex numbers?
A: No, this calculator is designed for real-valued functions of real variables.

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