Tangent line approximation

Definition

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.

5 Must Know Facts For Your Next Test

1. The formula for the tangent line approximation at $x=a$ is given by $L(x) = f(a) + f'(a)(x - a)$.
2. It provides an accurate estimate if the function is differentiable at the point $a$ and within close proximity to it.
3. Tangent line approximations are also known as linear approximations or local linearizations.
4. The method assumes that over small intervals, the function can be closely approximated by its tangent line.
5. Errors in approximation tend to increase as you move further from the point of tangency.

Review Questions

• What is the formula for the tangent line approximation at a point $x=a$?
• Why does a tangent line provide a good approximation near the point of tangency?
• How does the accuracy of a tangent line approximation change as you move away from the point of tangency?

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.