Answer:
a) approximate: 4.9400; calculator: 4.9396
b) approximate: 7.8800; calculator: 7.8806
Step-by-step explanation:
The linear approximation to a function f(x) can be written as ...
f(x+∆x) ≈ f(x) +(∆x)f'(x)
a) f(x) = √x; x = 25; ∆x = -0.6; f'(x) = 1/(2√x)
f(x) = 5; f'(x) = 1/(2·5) = 1/10
f(24.4) ≈ 5 + (-0.6)(1/10) = 4.9400 . . . approximation
f(24.4) = √24.4 ≈ 4.9396 . . . by calculator
The approximation is about 0.0074% high, a negligible amount for many applications.
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b) f(x) = x^3; x = 2; ∆x = -0.01; f'(x) = 3x^2
f(2) = 8; f'(2) = 12
f(1.99) ≈ 8 + (-0.01)(12) = 7.8800 . . . approximation
f(1.99) = 7.8806 . . . by calculator
The approximation is about 0.0076% low, a negligible amount for many applications.
