Calculus practice
Horizontal Tangents worksheets
Find every exact coordinate where polynomial, rational-polynomial, and trigonometric functions have horizontal tangent lines.
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Differentiate, solve f'(x) = 0, check the original domain, and report every exact point (x, f(x)).
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Customize this worksheetFind every coordinate, exactly
Each worked key solves the full derivative equation and substitutes every valid stationary input into the original function.
Learning objective
What this activity builds
Solve f'(x) = 0 completely, enforce the original function domain, and substitute every valid input into f for exact coordinates.
Worked example
Find the horizontal tangent points of f(x) = x^3 - 3x.
(-1, 2) and (1, -2).
f'(x) = 3x^2 - 3 = 0 gives x = -1 and x = 1. Substituting both inputs into the original function gives the two coordinates.
Teaching notes
Make the practice count
- Solve the derivative equation completely before substituting.
- Check holes and trigonometric poles in the original function.
- Keep radical coordinates exact and include stationary inflection points.
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Choose 6, 8, or 10 problems and any nonempty selection of polynomial, rational-polynomial, and trigonometric functions.