Math
Tangent lines.
Definition
A tangent line is a straight line that touches a curve at exactly one point and has the same slope as the curve at that point. In calculus, the slope of the tangent line at a point equals the derivative of the function evaluated at that point.
How it works · 4 phases
Step by step.
- Find the derivative f'(x) of the function.
- Evaluate f'(a) at the point of tangency x = a to get the slope.
- Find the y-coordinate by evaluating f(a).
- Write the equation using point-slope form: y − f(a) = f'(a)(x − a).
Examples
Real-world.
- 1 The tangent line to y = x² at x = 3 has slope 6, giving y = 6x − 9
- 2 Finding the tangent line to determine instantaneous velocity on a position graph
- 3 Using a tangent line to approximate √4.1 via linearization
Key Fact
Tangent line at x = a: y − f(a) = f'(a)(x − a)
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