Find the family of curves given the slope of the tangent line at any point by solving a separable differential equation.

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Finding a Family of Curves from Tangent Slopes

Antiderivatives core

Find the family of curves for which the slope of the tangent line at \( (x, y) \) is \( (1+x)/(1-y) \).

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What you need first

the geometric interpretation of the derivative
solving separable differential equations
completing the square
the standard equation of a circle

What this problem tests

Tests the ability to translate a geometric property into a differential equation, solve it by separation of variables, and identify the resulting conic section.

This problem is typical in introductory calculus courses when introducing differential equations and their geometric interpretations.

Finding a Family of Curves from Tangent Slopes

What you need first

What this problem tests

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