Replacing curves with lines

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Two young mathematicians discuss linear approximation.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Hmmmm. Riley, I just thought of something…
Riley: What is it?
Devyn: When we compute derivatives, we are looking at the slope of tangent lines right?
Riley: You know it.
Devyn: Well, I wonder: Instead of studying curves, could we just study “zoomed-in” lines?
Riley: I’m not sure…

You read someplace that

\[ \ell (x) = \frac {1}{4}(x-4)+2 \]

is a good approximation for $f(x) = \sqrt {x}$ when $x$ is close to $4$.

Plot $\ell (x)$ and $f(x)$. Explain how this shows that $\ell (x)$ is a good approximation when $x$ is close to $4$.

Explain (if you can) using concepts of calculus to explain why $\ell (x)$ is a good approximation for $f(x)$ when $x$ is close to $4$.

2025-01-06 20:03:11