First slide

Differentials; errors and approximations

Question

$The approximate value of f (5.001), where f (x) = x^{3} - 7 x^{2} + 15 .$

Moderate

A

-34.995
B

34.995
C

-38.458
D

38.458

Solution

$Let x = 5 and Δ x = 0.001 . Then, we have$
$\begin{array}{l} f (5.001) = f (x + Δ x) = (x + Δ x)^{3} - 7 (x + Δ x)^{2} + 15 \\ Now, Δ y = f (x + Δ x) - f (x) \end{array}$
$\begin{array}{l} ∴ f (x + Δ x) = f (x) + Δ y \\ Or \approx f (x) + f^{'} (x) \cdot Δ x (as d x = Δ x) \\ Or f (5.001) = [(5)^{3} - 7 (5)^{2} + 15] + [3 (5)^{2} - 14 (5)] (0.001) \\ = (125 - 175 + 15) + (75 - 70) (0.001) \\ = - 35 + 0.005 \\ = - 34.995 \end{array}$

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