Linear Approximation Calculator

This linear approximation calculator builds the linearization \( L(x) \) of a function at a point \( a \), uses it to estimate \( f(x) \) at a nearby value, and compares the estimate with the true value so you can see the error. It covers the standard “use a linear approximation to estimate…” problems from Calculus 1.

It’s free, instant and needs no sign-up.

How to use the linear approximation calculator

1. Enter the function \( f(x) \). Use ^ for powers, sqrt(x) for square roots, cbrt(x) for cube roots, sin(x), ln(x), e^(x) and pi. Implicit multiplication like 2x works.
2. Enter the center \( a \): a nearby point where \( f(a) \) is easy to compute, such as a perfect square for sqrt(x) or 1 for ln(x).
3. Enter the value \( x \) you want to estimate.
4. Press Linearize. The output shows \( L(x) \) in slope-intercept form, the estimate \( L(x) \), the actual \( f(x) \) and the absolute error \( |f(x) - L(x)| \).

Formula

The linearization of \( f \) at \( x = a \) is its tangent line:

$$L(x) = f(a) + f'(a)(x - a)$$

For \( x \) close to \( a \), \( f(x) \approx L(x) \). In differential notation, the change in \( f \) is approximated by \( dy = f'(a)\,dx \) with \( dx = x - a \). The calculator differentiates \( f \) exactly, evaluates \( f(a) \) and \( f'(a) \), and simplifies \( L(x) \) to \( mx + b \) form.

Worked example

Example 1. Estimate \( \sqrt[3]{8.3} \).

Take \( f(x) = \sqrt[3]{x} \) and \( a = 8 \), since \( \sqrt[3]{8} = 2 \).

• \( f'(x) = \frac{1}{3x^{2/3}} \), so \( f'(8) = \frac{1}{3\cdot4} = \frac{1}{12} \).
• \( L(x) = 2 + \frac{1}{12}(x - 8) = \frac{x}{12} + \frac{4}{3} \).
• \( L(8.3) = 2 + \frac{0.3}{12} = 2.025 \).

The true value is \( \sqrt[3]{8.3} \approx 2.0246939 \), so the error is about 0.00031. That’s the default example in the calculator above.

Example 2. Estimate \( \ln(0.9) \) using \( a = 1 \). Here \( f(1) = 0 \) and \( f'(1) = 1 \), so \( L(x) = x - 1 \) and \( L(0.9) = -0.1 \). The true value is \( \ln 0.9 \approx -0.1053605 \), an error of about 0.0054. Type ln(x), 1 and 0.9 to check.

Over- or underestimate? Reading the error

The output gives the size of the error, not its sign. Concavity tells you the direction:

• If \( f''(x) < 0 \) near \( a \) (concave down), the tangent line lies above the curve, so \( L(x) \) is an overestimate. Both examples are like this: \( \sqrt[3]{x} \) and \( \ln x \) are concave down for \( x > 0 \), and indeed \( 2.025 > 2.0246939 \) and \( -0.1 > -0.1053605 \).
• If \( f''(x) > 0 \) (concave up), \( L(x) \) is an underestimate.

Two more tips:

• Stay close to \( a \). The error grows roughly with \( (x - a)^2 \). Estimating \( \sqrt[3]{9} \) from \( a = 8 \) is fine; estimating \( \sqrt[3]{20} \) from \( a = 8 \) is not. Choose the nearest “nice” center instead (27 would be better there).
• Angles are in radians. For sin(x) or cos(x), enter \( a \) and \( x \) in radians, for example pi/6, not 30.

If you need more accuracy than a straight line gives, add a quadratic term; that’s the idea behind the Taylor series, and the Taylor series calculator builds those higher-degree polynomials. The by-hand method and a table of standard linearizations are in the linear approximation guide.

Related guides

• Linear Approximation (Linearization)
• How to Find the Equation of a Tangent Line
• Derivative of ln(x)
• Taylor and Maclaurin Series

Further reading

• Linear Approximations (Paul’s Online Math Notes) — the tangent line as an estimate, and why accuracy drops away from \( a \).
• Differentials (Paul’s Online Math Notes) — the \( dy = f'(x)\,dx \) notation and estimating measurement error.

FAQ

What is linear approximation used for?

It estimates function values that are hard to compute directly, such as roots and logarithms of numbers near “nice” values. It also underlies differentials, error propagation and Newton’s method.

Is linearization the same as the tangent line?

Yes. The linearization \( L(x) \) is the equation of the tangent line at \( x = a \), used as an approximation for \( f(x) \). The tangent line calculator gives the same line, along with the normal line.

How accurate is a linear approximation?

It is very accurate close to \( a \) and gets worse as you move away. The calculator shows the exact error for your values so you can judge.

Why is my estimate bigger than the real value?

Your function is concave down near \( a \), so the tangent line lies above the graph. For concave up functions the estimate comes out too small.