Question

SCALCET8 3.10.025. Use a linear approximation (or differentials) to estimate the given number. (Round your answer to five decimal places.) 3 126

Answer 1

`f(126) \approx 5.01333`

Explanation:

Given

`\sqrt[3]{126}`

Required

Solve using differentials

In differentiation:

`f(x+\triangle x) \approx f(x) + \triangle x \cdot f'(x)`

Express 126 as 125 + 1;

i.e.

`x = 125; \triangle x = 1`

So, we have:

`f(125+1) \approx f(125) + 1 \cdot f'(125)`

`f(126) \approx f(125) + 1 \cdot f'(125)`

To calculate f(125), we have:

`f(x) = \sqrt[3]{x}`

`f(125) = \sqrt[3]{125}`

`f(125) = 5`

So:

`f(126) \approx f(125) + 1 \cdot f'(125)`

`f(126) \approx 5 + 1 \cdot f'(125)`

`f(126) \approx 5 + f'(125)`

Also:

`f(x) = \sqrt[3]{x}`

Rewrite as:

`f(x) = x^(1)/(3)`

Differentiate

`f'(x) = (1)/(3)x^{(1)/(3) - 1}\\`

Using law of indices, we have:

`f'(x) = (x^(1)/(3))/(3x)`

So:

`f'(125) = (125^(1)/(3))/(3*125)`

`f'(125) = (5)/(375)`

`f'(125) = (1)/(75)`

So, we have:

`f(126) \approx 5 + f'(125)`

`f(126) \approx 5 + (1)/(75)`

`f(126) \approx 5 + 0.01333`

`f(126) \approx 5.01333`