Use derivatives to sketch the graph of a function

by RoRi

September 30, 2015

Let

$f(x) = x - \sin x.$

Find all points such that $f'(x) = 0$ ;
Determine the intervals on which $f$ is monotonic by examining the sign of $f'$ ;
Determine the intervals on which $f'$ is monotonic by examining the sign of $f''$ ;
Sketch the graph of $f$ .

We take the derivative,
$f'(x) = 1 - \cos x.$

Thus,

$f'(x) = 0 \quad \implies \quad 1 - \cos x = 0 \quad \implies \quad x = 2n \pi.$
$f$ is increasing for all $x$ since $1 - \cos x \geq 0$ for all $x$ .
Taking the second derivative,
$f''(x) = \sin x.$

Thus, $f'$ is increasing for $2 n \pi < x < (2n+1) \pi$ and decreasing for $(2n-1) \pi < x < 2n \pi$ .
We sketch the curve,

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One comment

November 1, 2020 at 2:52 am

Jan says:

You get a better understanding of this function if you increase the domain to e.g. (-20,20).

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