Half-width substitution

Geometric meaning

Equivalent expression

Let {f}(x) be differentiable on the interval I

{f}' (x) \ge 0 ( \le 0)

Let {f} (x) be differentiable on the interval \left ( a,b \right)

For all x \in \left ( a,b \right ) , there is {f}' (x) \ge 0 ( \le 0)

{f}'(x) e 0 on any self-interval of \left ( a,b \right )

This judgment also holds true if the function is one-sided and continuous on the closed side of the interval.

\frac{0}{0} type limit

\frac{a}{\infin} type infinitive limit

LawLópidaLaw#Imitation

Define derivative

{f} (x)= \sum_{i=0}^{n} \frac{{f}^{(i)}(0)}{n!} (x)^i

{T} (x) =\sum_{i=0}^{n} \frac{{f}^{(i)}(x_0)}{n!} (x-x_0)^i \frac{f^ {(n 1)}(\xi)}{(n 1)!} (x-x_0)^{n 1}

Let {f} be continuous at point x_0 and differentiable on a certain neighborhood U ^ {\circ} (x_0; \delta)

(i) If {f} '(x) \le 0 when x \in \left ( x_0 - \xi ,x_0 \right ), {f when x \in \left( x_0 , x_0 \xi \right ) }' (x) \ge 0, then {f} obtains the minimum value at x_0

(ii) If when x \in \left ( x_0 - \xi ,x_0 \right ) {f} '(x) \ge 0, when x \in \left( x_0 , x_0 \xi \right ) {f }' (x) \le 0, then {f} obtains the maximum value at x_0

Suppose f is first-order differentiable on a certain neighborhood U (x_0; \delta) of x_0, and second-order differentiable at x=x_0, and {f} '(x_0)= 0, {f} '' (x_0) e 0

If {f}''(x_0) < 0, then {f} obtains the maximum value at x_0

If {f}''(x_0) > 0, then {f} obtains the minimum value at x_0

Suppose {f} exists in a certain neighborhood of x_0 with derivatives up to order n-1, and is derivable to order n at x_0, and {f} ^ {(k)} (x_0) = 0 (k=1,2,\dots ,n-1), {f}^{(n)} e 0

When n is an even number, {f} takes the extreme value at x_0

When n is an odd number, {f} does not take an extreme value at x_0

The three sufficient conditions do not apply to determine all extreme points (even if they are differentiable)

The maximum point may not have a left (right) neighborhood that makes it monotonic.

Lemma f is the necessary and sufficient condition for the convex function on I

Theorem Suppose f is a differentiable function on the interval I, then the following statements are equivalent to each other

The necessary and sufficient condition for the minimum value of a differentiable convex function is that the derivative is zero

The convex function on the open interval does not take the maximum value

Formula Jensen (Jensen) inequality

A convex function on the open interval I has left and right derivatives at any point on I

{f} is a convex function on the open interval I, then {f} is bounded on any closed subinterval \left [ a, b \right ] of I

Convert rational numbers to infinite decimals for comparison

size

∞

-∞

The sum and difference product of two infinitesimal quantities is still an infinitesimal quantity

The product of an infinitesimal quantity and a bounded quantity is an infinitesimal quantity

High level/low level

Same level

equivalence

Discontinuities of the first kind

Type II discontinuities

boundedness theorem

{f}'( x_{0} ) =\lim _ { x \to x _ { 0 } } \frac { f ( x ) - f ( x _ { 0 } ) } { x - x _ { 0 } } =\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}=\lim_{\Delta x\to 0} \frac{f(x_{0} \Delta x)-f(x_{0})}{\Delta x}

Definition is not derivable

formula finite increment formula

Theorem is differentiable\Rightarrow continuous (but not vice versa)

Conditions for the existence of theorem {f}'(x_0)

Define stable point

Fermat's theorem

Corollary If the function {f} is differentiable on the interval I, and {f}' (x) = 0, x \in I, then {f} is a constant function on I

Corollary If the functions {f} and {g} are both differentiable on the interval I, and {f} ' (x) = {g} ' (x) , x \in I, then on the interval I, {f} ( x) ={g} (x) c (c is a constant)

Corollary Theorem Derivative Limit Theorem

(u \pm v) '=u ' \pm v '

(uv) '=u 'v v 'u

(\frac{u}{v}) '=\frac{u 'v-v 'u}{v^2}

f '(x_0)=\frac{1}{f^{-1}(y_0)}

\frac{\mathrm{d} y}{\mathrm{d} x} =\frac{1}{\frac{\mathrm{d} y}{\mathrm{d} x} }

({f}\circ {\varphi}) '(x_0)={f '}(u_0){\varphi} '(x_0)

\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} u}\cdot \frac{\mathrm{d} u}{ \mathrm{d} x}

(\sin x) '=\cos x

(\cos x) '=-\sin x

(\tan x) '=\sec^2 x

(\cot x)'=-\csc ^2 x

(\sec x) '=\sec x \tan x

(\csc x) '=-\csc x \cot x

(e^x) '=e^x

(\ln x) '=\frac{1}{x}

[{u} \pm {v} ]^{(n)}={u}^{(n)} \pm {v}^{(n)}

Formula Leibniz formula

({u}{v})^{(n)}= \sum_{k=0}^{n} {C_{n}^{k} {u}^{(n-k)}{v}V^{ (k)}}

where {u}^{(0)}={u},{v}^{(0)}={v}

Invariance of first-order differential forms

To replace the song directly

Error limit of measured value x_0\delta _x \ge |x-x_0|=|\Delta x|

| \ Delta y | = | {f} (x) -{f} (x_0) | \ approx | {f} '(x_0) \ delta x | \ le | {f}' (x_0) | \ delta_x

Relative error limit\frac{ \delta_y}{|y_0|}=|\frac{{f} '(x_0)}{{f}(x_0)}| \delta_x