Linearization is one of 1 approximation formula in the calculus section of this library, and it is used at ap · university level.
Why linearization works
Zoom in far enough on a differentiable curve and it becomes indistinguishable from its tangent line. So starting at the known height f(a) and walking along the tangent slope for a distance of x − a lands very close to the curve, as long as you do not walk far.
What each symbol means
$a$ is the center and $L$ is the tangent-line approximation.
Linearization: when it holds
$f$ must be differentiable near $a$; accuracy usually decreases farther from $a$.
When it stops applying
Far from the centre the tangent drifts away from the curve, and it drifts fastest where the graph bends hardest. Estimating the square root of 101 from a centre of 100 is fine; using that same tangent line at x = 4 predicts 5.2 when the true root is 2.
Linearization: a worked example
$\sqrt{4.1}\approx2+\frac14(0.1)=2.025$.
The mistake to avoid
What people do: Writing the second term as the slope times x instead of the slope times x − a.
Why it goes wrong: That measures the walk from 0 rather than from the centre a, so the estimate is off by the slope multiplied by a, which is usually a large error.
Do this instead: Always subtract the centre. The second term is the slope at a times the distance you moved away from a.
Linearization: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is the center and $L$ is the tangent-line approximation.
- Check the conditions before substituting. $f$ must be differentiable near $a$; accuracy usually decreases farther from $a$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Approximation
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where linearization comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about linearization
How do I choose the centre a?
Take the nearest input whose exact value you already know, such as a perfect square or a familiar angle. The closer the centre, the smaller the error.
Does the estimate come out too high or too low?
It sits above the curve wherever the graph is concave down and below it wherever the graph is concave up, because the tangent line lies on the outside of the bend.
How far off is the estimate 2.025 for the square root of 4.1?
The true value is about 2.0248457, so the tangent line is high by roughly 0.00015, which is about one part in thirteen thousand.
Is this the same thing as a differential?
Almost. The differential reports only the change predicted by the slope, while linearization adds the starting height back on to give an actual estimated value.