Differentials or linearization

[golflim]

Use differentials or linearization to find an approximate value of the square root of 35. Is the answer an overestimate or underestimate?Explain why.

Any help on this would be great thanks

 

[tkhunny]

f(x) = sqrt(x)
df(x,dx) = (1/[2*sqrt(x)])*dx

Is is known that f(36) = sqrt(36) = 6
What is sqrt(35)?

f(35) = f(36-1) = apprx(f(36) + df(36,(35-36))) = 6 + [1/2*sqrt(36)]*(-1) = 6 - 1/12 = 5.9167

The second derivative (you calculate that) is negative on [35,36], so it is an overestimate.

5.9167² = 35.00733889 Yup. Too much. With more decimal places, that's 35.00694..., still too much.