Use linear approximation (the tangent-line estimate) to find $2.01$ correct to four decimal places. Write $f (x) = x$ and take the base point as $a = 2$ .

easy 3 marks Class 12 · Mathematics · Application of Derivatives

Answer

2.01 \approx 1.4142

(to 4 d.p.)

The correct answer is $2.01 \approx 1.4142$ (to 4 d.p.) — Step 5 – Round to four decimal places: $1.4177$ . (A calculator confirms $2.01 \approx 1.41774 \dots$ , so the estimate is correct to four d.p.)

Step-by-step solution

Step 1 – Choose the function and the nearby 'nice' point. We want $2.01$ , so set $f (x) = x$ and pick $a = 2$ because $2$ is a well-known surd and $2.01$ is very close to $2$ .
Step 2 – Evaluate $f (a)$ and $f^{'} (a)$ . $f (2) = 2 \approx 1.41421$ . Differentiate: $f^{'} (x) = \frac{1}{2 x}$ , so $f^{'} (2) = \frac{1}{2 2} \approx 0.35355$ .
Step 3 – Identify the small increment. $h = 2.01 - 2 = 0.01$ .
Step 4 – Apply the linear-approximation formula $f (a + h) \approx f (a) + h f^{'} (a)$ . Substitute: $2.01 \approx 1.41421 + 0.01 \times 0.35355 = 1.41421 + 0.0035355 = 1.4177455$ .
Step 5 – Round to four decimal places: $1.4177$ . (A calculator confirms $2.01 \approx 1.41774 \dots$ , so the estimate is correct to four d.p.)

Common mistake: Students often differentiate

x

1

instead of

\frac{1}{2 x}

, or they forget to evaluate

f^{'} (a)

at the chosen base point and substitute an incorrect slope.

Concept tested: Linear approximation using the first-order Taylor expansion $f(a+h)\approx f(a)+h\,f'(a)$

Published 19 September 2026 · How this solution was prepared and checked

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Use linear approximation (the tangent-line estimate) to find 2.01​ correct to four decimal places. Write f(x)=x​ and take the base point as a=2.

Step-by-step solution

Use linear approximation (the tangent-line estimate) to find $2.01$ correct to four decimal places. Write $f (x) = x$ and take the base point as $a = 2$ .