7. Use linear approximation to approximate the value of 63 without the need for a calculator.
7. Use linear approximation to approximate the value of 63 without the need for a calculator.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:7. Use linear approximation to approximate the value of 63 without the need for a calculator.
Expert Solution
Step 1
We can use linear approximation to approximate the value of ³√63.
First, let's choose a point close to ³√63 that we know the exact cube root of. The easiest choice is to use 8, since 8³ = 512, and we know that ³√512 = 8.
Next, we can use the linear approximation formula to find an approximate value of ³√63 based on the tangent line at x = 8.
The linear approximation formula is given by:
f(x) ≈ f(a) + f'(a)(x-a)
where a is the point we're approximating around, f'(a) is the derivative of f at a, and x is the point we want to approximate.
In this case, we have:
f(x) = ³√x, so f(8) = 2
f'(x) = 1/(3x²), so f'(8) = 1/192
x = 63
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