$f'\to f$

How to go from $f'(x)$ to $f(x)$? Key concept:

$f'(x)$ is the slope of $f(x)$.

This is the same problem as... $$f \to F$$

because, when $\int_a^x f(t)\,dt=F(x)-F(a)$, then $f(x)=F'(x)$. When the problem is phrased like this, $f(x)$ is the slope of $F(x)$ is $f(x)=F'(x)$ and the key concept

$$F(x)=F(a)+\int_a^x F'(t)\,dt$$

Identify where $f'(x)$ is +, -, 0

Identify $x$ values where $f'(x)$ is positive: $f'(x)$ is above the $x$-axis. Ignore the slope of $f'$!

Identify $x$-values where $f'(x)$ is negative, and zero.

Layout the outline of $f(x)$

$f'(x)$ represents the slope of $f(x)$. Sketch a very rough $f(x)$ which has a + or - slope, depending on the $x$ values you identified.

If I haven't told you the starting $y$ value, you can pick *any* $y$ value to start with.

Identify critical points of $f(x)$

Where $f'(x)$ crosses the $x$-axis, the slope of $f(x)$ is zero: a critical point.

Where $f'(x)$ crosses the $x$-axis, the slope of $f(x)$ is zero: a critical point.

Identify inflection points of $f(x)$

Identify $x$ values where the slope of $f'$ is zero (that is to say, $(f')'=0$).

These are $x$ values where $f(x)$ has an inflection point: Where it changes concavity.

Examples

Examples

Examples

Examples

Examples