Tangent line
The straight line that touches a curve at one point and has the same slope there. Its equation is y - f(a) = f'(a)(x - a).
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Analogy
A ruler laid against a bend in the road, touching it at just one spot.
Example
f(3) = 9 and f'(3) = 6.y - 9 = 6(x - 3).So y = 6x - 9.At x = 3.1 the line gives 9.6; true value 9.61.
In practice
Tangent lines give quick, good estimates of a function near a point.
Practice
Interview questions
- Find the equation of the tangent line to y = x^2 at x = 3, and explain what the derivative represents.
- f(3) = 9 gives the point (3, 9)
- f'(x) = 2x so the slope is 6
- Tangent line y - 9 = 6(x - 3), so y = 6x - 9
- Derivative is the instantaneous rate of change
Questions people ask
How do I find the equation of a tangent line?
Find the point (a, f(a)) and the slope f'(a). Then write y - f(a) = f'(a)(x - a) and tidy up. For y = x squared at x = 3, the point is (3, 9) and the slope is 6, giving y = 6x - 9. Check: x = 3 gives back 9.
What is a normal line?
The normal line passes through the same point as the tangent but at right angles to it. Its slope is the negative reciprocal of the tangent's slope. If the tangent slope is 6, the normal slope is -1/6. Normal lines matter in physics, for example when finding the direction a surface pushes back on an object resting on it.