Question: "Use a linear approximation (or differentials) to estimate..."

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2021-10-08

Use a linear approximation (or differentials) to estimate the given number. (1.999)4

Arham Warner

Skilled2021-10-09Added 102 answers

Step 1

We know that lim_{△ x \to 0} \frac{f (x + △ x) - f (x)}{△ x} = f^{'} (x)

Hence for small values of △ x; we can say that

\frac{f (x + △ x) - f (x)}{△ x} \approx f^{'} (x)

f (x + △ x) \approx f (x) + △ x \cdot f^{'} (x)

Let f (x) = x^{4}, △ x = - 0.001, x = 2

Then we have

f (2 - 0.002) \approx f (2) + (- 0.001) \cdot f^{'} (2)

{(1.199)}^{4} \approx 2^{4} + (- 0.001) \cdot f^{'} (2) \to (1)

The only Unknown we have is f^{'} (2)

Step 2

Finding f^{'} (2)

f (x) = x^{4}

Differentiate

f^{'} (x) = 4 x^{3}

f^{'} (2) = 4 {(2)}^{3} = 32

Step 3

Substitute the value of f^{'} (2) in Enq(1)

{(1.999)}^{4} \approx 2^{4} + (- 0.001) \cdot (32)

{(1.999)}^{4} \approx 16 - 0.032 = 15.968

Result

{(1.999)}^{4} \approx 15.968

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