Approximation Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Use the first-order linear approximation a+ฯตโ‰ˆa+ฯต2a\sqrt{a+\epsilon} \approx \sqrt{a} + \dfrac{\epsilon}{2\sqrt{a}} to estimate 101\sqrt{101}.

Solution

  1. 1
    Write 101=100+1\sqrt{101} = \sqrt{100 + 1} with a=100a = 100 and ฯต=1\epsilon = 1.
  2. 2
    Apply the formula: 101โ‰ˆ100+12100=10+120=10.05\sqrt{101} \approx \sqrt{100} + \dfrac{1}{2\sqrt{100}} = 10 + \dfrac{1}{20} = 10.05.
  3. 3
    Compare: calculator gives 101โ‰ˆ10.0499\sqrt{101} \approx 10.0499, so the approximation is accurate to 44 decimal places.

Answer

101โ‰ˆ10.05\sqrt{101} \approx 10.05
The linear approximation formula uses the derivative of x\sqrt{x} at a nearby perfect square to estimate the shift. It is a powerful technique โ€” without a calculator, you can estimate many square roots to several decimal places using just arithmetic.

About Approximation

A value intentionally chosen to be close to but not exactly equal to the true value, with a known or estimated error.

Learn more about Approximation โ†’

More Approximation Examples

Example 1 easy

Estimate [formula] to one decimal place without a calculator.

Example 2 medium

Use the approximation [formula] to estimate the circumference and area of a circle with radius [form

Example 3 easy

Estimate [formula] using rounding, then compute the exact answer and find the percent error.

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