Substitution easy question

Hello, could anyone help me a bit? I don't know how I could determine that g' measures how much a small interval shrinks or stretches from that graph. And the step in derivation marked with red arrow.

Easy Math Editor

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Comments

Hi Nezajem123 Nezajem123, I believe you're referring to this problem.

the reasoning goes like this:

The (infinitesimal) change in $x$ ( $\Delta x$ ) is the difference between the images of the corresponding endpoints of the interval along the $u$ direction under $g$ . In short, $\Delta x = g(u_1)-g(u_0)$ with $\Delta u = u_1-u_0$ .

We want to get a relationship between $\Delta x$ and $\Delta u$ from this expression. We start by writing $u_1$ as $u_0+ \Delta u$ , which implies $\Delta x = g(u_0+\Delta u) - g(u_0)$ . Because $\Delta u$ is so very small, we can use a linear approximation to replace $g(u_0 +\Delta u )$ by $g(u_0) + g’(u_0) \Delta u$ . This implies that $\Delta x \approx g(u_0) + g’(u_0) \Delta u - g(u_0)$ , and so $\Delta x \approx g’(u_0) \Delta u$ . Another way of expressing this is as $\Delta x/ \Delta u = g’(u_0)$ , so $g’(u_0)$ measure how much $g$ stretches/shrinks small intervals along the $u$ -direction into small intervals in the $x$ -direction.

Hope that helps!

Brilliant Mathematics Staff - 1 year ago

Thank you very much. The step I didn't catch is linear approximation, another quiz solved it for me :) https://brilliant.org/practice/linear-approximation/?p=1

nezajem123 nezajem123 - 12 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$