Why is $y=f(a)+f'(a)(x-a)$ linear?

Question

I'm reading this book: Functions of Several Variables and I didn't understand why this function is linear:

I think the authors made a mistake.

Linear means degree 1. If 'a' is constant, then clearly it is. — Rishi, Dec 3 '16 at 12:26
This is linear with respect to $x$. Don't forget, $a$ is fixed! You can see that the slope of the line is $f'(a)$, and the y-intercept is $f(a)-af'(a)$, by expanding into the $y=mx+c$ form. — Teresa Lisbon, Dec 3 '16 at 12:26
@Rishi Right! I was thinking about the linear algebra definition of "linear" — user42912, Dec 3 '16 at 12:27
@астонвіллаолофмэллбэрг yes I know, I was thinking about the linear algebra definition of "linear" — user42912, Dec 3 '16 at 12:28

Community · Accepted Answer · 2017-04-13 12:21:00Z

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I think the author meant "the affine function $y=f(a)+f'(a)(x-a)$" instead of "the linear function". Please have a look at this.

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answered Dec 3 '16 at 12:29

Olivier Oloa

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$\begingroup$ Certainly he did. Worth the point out. $\endgroup$ – Teresa Lisbon Dec 3 '16 at 12:33
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$\begingroup$ It is common to refer to a function $f(x)=ax+b$ as linear, especially in high-school mathematics. $\endgroup$ – Wojowu Dec 3 '16 at 12:50
$\begingroup$ @Wojowu especially in calculus. I forgot that, shame on me. $\endgroup$ – user42912 Dec 3 '16 at 13:25
$\begingroup$ ...Or a linear polynomial. $\endgroup$ – The Vee Dec 3 '16 at 13:34

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