The approximate value of (3.978)^{frac{3}{2}} | vedclass

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Question

(A) Let $f(x) = x^{\frac{3}{2}}$.Then,$f'(x) = \frac{3}{2} x^{\frac{1}{2}}$.Using the linear approximation formula $f(a+h) \approx f(a) + h \cdot f'(a

Vedclass · Accepted Answer

$7.934$

Vedclass · Answer

(A) Let $f(x) = x^{\frac{3}{2}}$.Then,$f'(x) = \frac{3}{2} x^{\frac{1}{2}}$.Using the linear approximation formula $f(a+h) \approx f(a) + h \cdot f'(a)$,where $a = 4$ and $h = -0.022$:$f(4 - 0.022) \approx f(4) + (-0.022) \cdot f'(4)$.$f(4) = 4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^3 = 8$.$f'(4) = \frac{3}{2} \cdot 4^{\frac{1}{2}} = \frac{3}{2} \cdot 2 = 3$.Therefore,$f(3.978) \approx 8 + (-0.022) \cdot 3$.$f(3.978) \approx 8 - 0.066 = 7.934$.

The approximate value of $(3.978)^{\frac{3}{2}}$ is

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