Answer :
Let f(x) = sqrt(x). Use linear approximation about x = 25, since 24 is close to 25, and since it is easy to take the square root of 25.
Note that f ' (x) = 1/[2sqrt(x)] from the power rule.
About x = 25, f(x) is approximately f(25) + f ' (25)(x - 25).
So sqrt(24) = f(24) is approximately
f(25) + f ' (25)(24 - 25) = sqrt(25) + {1/[2sqrt(25)]}(-1) = 5 - 1/10 = 4.9 .
A linear approximation of f(x) in a neighborhood of a
f(x) ~ f(a) + f'(a) (x-a)
f(x) = x^1/4
2^4 = 16 i.e. a = 16
f(15) ~ f(16) + f'(16) (15-16)
f'(x) = 1/4 x^(-3/4)
f(15) ~ 2 + 1/4 (16)^-3/4 (-1)
~ 2 - 1/(32) = 1.96875
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f(x) ~ f(a) + f'(a) (x-a)
f(x) = x^1/4
2^4 = 16 i.e. a = 16
f(15) ~ f(16) + f'(16) (15-16)
f'(x) = 1/4 x^(-3/4)
f(15) ~ 2 + 1/4 (16)^-3/4 (-1)
~ 2 - 1/(32) = 1.96875
I hope my answer has come to your help. Thank you for posting your question here in Brainly. We hope to answer more of your questions and inquiries soon. Have a nice day ahead!