Question

Estimate the value of sqrt 99.8 by using a linear approximation

Answer 1

To find

√99.8 using linear approximation.

Solution:-

1. Let us consider, f(x)= √(100+x).  

2. Then we have, f'(x)= 1/(2√(100+x))

3. Now, f'(0)=$\lim_{x \to 0}\frac{f(x)-f(0)}{x}$  

But if x is very small,

f'(0) ≈ (f(x) - f(0))/x.

4. From 1 and 2, f(0)=10 and f'(0)=1/20.

5. Hence we get,  (1/20) ≈ (f(x) - 10)/x

6.  ∴  f(x) ≈  (10) + x/20

7.  To find √99.8 = √100 - 0.2  = f(-0.2).

8. Substituting x=-0.2 in (6),

9. f (-0.2) ≈ 10 - 0.2/20 = 9.99

10. ∴ √99.8 ≈ 9.99.