Linear Approximation Calculator

Estimate function values near a point using tangent line linearization. Perfect for quick approximations and understanding differentials.

Linear approximation formula: L(x) = f(a) + f'(a)·(x - a)

For x close to a, f(x) ≈ L(x). The error is of order (x-a)².

Enter a function of x. Use ^ for exponent, * for multiplication, sqrt(), sin(), cos(), exp(), log() etc.
√x
sin(x)
eˣ
ln(x)
x³
1/(1+x)
tan(x)
∛x
The point where we know the exact value and derivative.
The x‑value at which to approximate f(x).
Range around expansion point for graph.
Results update automatically.
Computing linear approximation...

Understanding Linear Approximation

Linear approximation (or linearization) uses the tangent line at a known point to estimate the function's value at nearby points. It's the first‑order Taylor expansion and is widely used in science, engineering, and finance to simplify complex calculations.

Definition: For a differentiable function f, the linear approximation near x = a is:

L(x) = f(a) + f'(a)·(x - a)

The error (remainder) is approximately ½f''(c)(x-a)² for some c between a and x.

Common Linear Approximations (near 0)

Function Linearization L(x) Valid near
sin(x) ≈ x L(x)=x x=0
cos(x) ≈ 1 L(x)=1 x=0
eˣ ≈ 1 + x L(x)=1+x x=0
ln(1+x) ≈ x L(x)=x x=0
(1+x)ᵏ ≈ 1 + kx L(x)=1+kx x=0
√(1+x) ≈ 1 + x/2 L(x)=1+0.5x x=0

Applications

1
Physics: Approximating pendulum period for small angles (sinθ ≈ θ).
2
Economics: Marginal analysis – change in cost/revenue for small production changes.
3
Engineering: Linearizing non‑linear systems for control theory.
4
Numerical methods: Newton‑Raphson root finding uses linear approximation iteratively.

Calculator features:

  • Symbolic differentiation (nerdamer) for exact derivatives.
  • Instant comparison between approximation and actual value.
  • Absolute and relative error displayed.
  • Graph shows function, tangent, and both points of interest.
  • Handles trigonometric, exponential, logarithmic, and polynomial functions.

Frequently Asked Questions

It is most accurate when x is very close to a. The error is proportional to (x‑a)², so doubling the distance quadruples the error typically.

They are mathematically the same: the tangent line is used to approximate the function. Linear approximation is just the application of the tangent line to estimate function values.

This calculator is for single‑variable functions only. For multivariable linear approximation, you would need a gradient‑based calculator.

Absolute error = |f(x) - L(x)|. Relative error = |f(x)-L(x)| / |f(x)| (as percentage). It shows how good the approximation is.