Application of Derivatives Notes

🎓 Free mentorship — Join our community →🎓 Free mentorship — Join our community →🎓 Free mentorship — Join our community →🎓 Free mentorship — Join our community →

Library

Application of DerivativesGeneral introduction

Page 1/29 0/44 questions known0%

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

Solution continues

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

Solution continues

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

Solution continues

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

Solution continues

Solve it on paper, then

Solution continues

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

After the notes

Formula sheet

14 results from this chapter, grouped the way the notes teach them.

Monotonicity at a point

$f$ increasing at $x=a$: $f(a-h)<f(a)<f(a+h)$ for small $h>0$
$f$ decreasing at $x=a$: $f(a-h)>f(a)>f(a+h)$ for small $h>0$
Differentiable $f$ : $f'(a)>0$ / $f'(a)<0$: Increasing / decreasing at $x=a$
$f'(a)=0$ but $f$ still increasing or decreasing at $a$ (e.g. $x^3$ at $0$ ): Point of inflection

Monotonicity in an interval

Strictly increasing on $I$: $f'(x)\ge0$ on $I$ , with $f'(x)=0$ only at discrete points
Strictly decreasing on $I$: $f'(x)\le0$ on $I$ , with $f'(x)=0$ only at discrete points
Non-decreasing on $D$: $x_1>x_2\Rightarrow f(x_1)\ge f(x_2)$ ; i.e. $f'(x)\ge0$ , may be $0$ on an interval
Non-increasing on $D$: $x_1>x_2\Rightarrow f(x_1)\le f(x_2)$ ; i.e. $f'(x)\le0$ , may be $0$ on an interval
Critical points of $f$: Points in the domain where $f'(x)=0$ or $f'(x)$ does not exist

Composites and values

$f$ and $g$ both increasing (or both decreasing): $f\circ g$ is increasing
One of $f,g$ increasing, the other decreasing: $f\circ g$ is decreasing
Increasing $f$ on $[a,b]$ : least and greatest value: $f(a)$ and $f(b)$
Greatest / least value of $f$ on $[a,b]$: Largest / smallest of $f(a)$ , $f(b)$ and $f$ at the critical points in $(a,b)$
Proving $f(x)>g(x)$ for $x>a$: Let $h=f-g$ ; show $h(a)\ge0$ and $h'(x)>0$ for $x>a$

Last look

Quick revision

The checks to run before you sit a question on functions.

1Find intervals of monotonicity from the sign scheme of $f'(x)$ — mark every critical point, including where $f'$ does not exist.
2For "increasing for all $x\in\mathbb R$ ", allow $f'(x)=0$ at isolated points: solve $f'(x)\ge0$ , then check the boundary value separately.
3At a point where $f$ is not differentiable, compare $f(a-h)$ , $f(a)$ , $f(a+h)$ directly — the derivative test does not apply.
4A function increasing on two intervals need not be increasing on their union — check the jump at the joining point.
5"No critical point" means $f'(x)$ never equals $0$ : keep $a$ outside the range of the other side of $f'(x)=0$ .
6Range of a continuous function: monotonic pieces + values at critical points + limits at the open ends.
7Number of solutions of $f(x)=k$ : a strictly monotonic continuous $f$ takes each value in its range exactly once.
8For inequalities, move everything to one side, differentiate, and use the value at the starting point.
9Comparing $\pi^e$ and $e^\pi$ : study $x^{1/x}$ (or $\frac{\ln x}{x}$ ), which decreases for $x>e$ .