Use a derivative to find a tangent line
The idea
Evaluate the original function at the chosen x-coordinate to obtain a point on the curve.
Step by step
- Evaluate the original function at the chosen x-coordinate to obtain a point on the curve.
- Differentiate, then evaluate the derivative at that x-coordinate to find the tangent slope.
- Use y − f(a) = f′(a)(x − a), or substitute the point into y = mx + b, and check the line passes through it.
Example
For f(x) = x2 + 2, the tangent line at x = 1 is y = 2x + b. What is b?
Answer
1
The derivative gives slope 2x, or 2 at x = 1. The tangent passes through (1, 3); b = 3 − (2)(1) = 1.
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For f(x) = x2 + 1 at x = 2, find the point, slope and tangent equation. Check the point in both equations.
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