Mathematical foundation
Linearization L(x)=f(a)+f′(a)(x−a) is the tangent-line approximation near a. Linear Approximation can be checked against line equation.
The operation is a compact expression of the linear approximation definition above.
Math calculator
Linear Approximation Calculator
Approximate f(x) near a base point with its tangent line. Input changes update both linear estimate and the supporting steps.
Linear Approximation inputs
Complete the fields
Steps without the calculator
Compute the base value and derivative, multiply slope by displacement, and add.
A worked example
For √x near 4, L(4.1)=2+(1/4)(0.1)=2.025.
Linear Approximation in practice
It estimates roots, measurements, and small input changes without a full reevaluation. A related application of Linear Approximation is differential change.
Numerical limits of Linear Approximation
Accuracy generally deteriorates as the target moves farther from a or curvature grows.
When linear approximation looks surprising, restore the sample and vary base point a by itself.
Checking the model and magnitude
For this linear approximation calculation, the labels Function f(x), Base point a and Nearby target x carry mathematical meaning. A transposed entry can remain numerically valid while describing an entirely different setup.
How the output responds
Changing target distance scales the first-order correction directly. A small controlled input change is enough to test the expected direction.
Precision and reporting
An exact expression can preserve factors, radicals, or π that a decimal hides. Use the representation suited to the next task and label it clearly as linear estimate.
Differential approximation reports the change dy and updated value from dx. Checking the requested noun is often enough to select the right model.
Reproducibility here depends on the inputs more than the interface. Preserve Function f(x), Base point a and Nearby target x, the operation shown, and enough unrounded digits for the next calculation.
Reproducing this result later
Keep Function f(x), Base point a, and Nearby target x with the Linear Approximation output. Preserve more digits internally than the final report needs. Subtraction in finite differences and cumulative integration can lose significant digits, so rounding intermediate samples can damage the final estimate disproportionately.
For this linear approximation result, treat the shown digits as conditional on the entered domain and sampling choices. Keeping those choices with the value helps distinguish a stable computation from a coincidental rounded match. Recheck the actual function value when it is available and report the approximation error beside the local estimate.
Questions about Linear Approximation
Is this exact?
Only for linear functions.
What makes it accurate?
A nearby target and modest curvature.
What is L(a)?
Exactly f(a).