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Linearization — Definition, Formula & Examples

Linearization is the process of approximating a function near a specific point by using the tangent line at that point. The tangent line becomes a simple linear function that closely matches the original curve for values of xx near the point of tangency.

Given a function ff differentiable at x=ax = a, the linearization of ff at aa is the function L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a). This is the first-degree Taylor polynomial of ff centered at aa, and it provides the best linear approximation to ff near aa.

Key Formula

L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a)
Where:
  • L(x)L(x) = The linear approximation of f at values near a
  • f(a)f(a) = The value of the function at the center point
  • f′(a)f'(a) = The derivative of f evaluated at a (slope of the tangent line)
  • aa = The center point where the tangent line touches the curve

How It Works

To linearize a function, pick the point x=ax = a where you want the approximation centered. Evaluate both f(a)f(a) and f′(a)f'(a). Substitute these values into L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a). You can then estimate f(x)f(x) for any xx close to aa by computing L(x)L(x) instead. The approximation improves the closer xx is to aa, and it works best when the function does not curve sharply near that point.

Worked Example

Problem: Use linearization to approximate 4.1\sqrt{4.1} without a calculator.
Choose a center point: Pick a=4a = 4 because 4\sqrt{4} is easy to compute and 4.1 is close to 4.
f(x)=x,a=4f(x) = \sqrt{x}, \quad a = 4
Evaluate f(a) and f'(a): Find the function value and derivative at the center point.
f(4)=2,f′(x)=12x,f′(4)=14f(4) = 2, \quad f'(x) = \frac{1}{2\sqrt{x}}, \quad f'(4) = \frac{1}{4}
Build L(x) and approximate: Substitute into the linearization formula and evaluate at x=4.1x = 4.1.
L(x)=2+14(x−4)  ⟹  L(4.1)=2+14(0.1)=2.025L(x) = 2 + \tfrac{1}{4}(x - 4) \implies L(4.1) = 2 + \tfrac{1}{4}(0.1) = 2.025
Answer: 4.1≈2.025\sqrt{4.1} \approx 2.025. The actual value is 2.02485…2.02485\ldots, so the linearization is accurate to about four decimal places.

Why It Matters

Linearization is a central technique in AP Calculus AB Unit 4, where you must approximate function values using local linearity. Engineers and scientists rely on linearization to simplify complex models—for instance, small-angle approximations in physics replace sin⁡θ\sin\theta with θ\theta, which is exactly the linearization of sine at θ=0\theta = 0.

Common Mistakes

Mistake: Using a center point aa that is far from the target xx value.
Correction: Always choose aa as close to the target as possible while still being a value where f(a)f(a) and f′(a)f'(a) are easy to compute. The farther xx is from aa, the less accurate the approximation becomes.

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