A tangent line approximation uses the tangent line at a point to estimate the value of a function near that point. It is based on linearization and is useful for making quick, approximate calculations.
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The formula for the tangent line approximation at $x=a$ is given by $L(x) = f(a) + f'(a)(x - a)$.
It provides an accurate estimate if the function is differentiable at the point $a$ and within close proximity to it.
Tangent line approximations are also known as linear approximations or local linearizations.
The method assumes that over small intervals, the function can be closely approximated by its tangent line.
Errors in approximation tend to increase as you move further from the point of tangency.
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Related terms
Differentiable: A function is differentiable at a point if it has a defined derivative there, meaning it can be locally approximated by a linear function.
Linearization: The process of approximating a function near a given point using its tangent line.
$f'(a)$: The derivative of the function $f(x)$ evaluated at $x=a$, representing the slope of the tangent line at that point.