微积分笔记

30% for quizzes

60% for tests (T1 25%, T2 25%)

4 credits = 200 min in a week

minimum 400 min <- out of dats in a week.

The most important rule is – WHEN A SENTENCE/ WORD IS NOT UNDERSTAND, STOP ME AND ASK TO EXPLAIN AGAIN.

“The explain is my job, to understand is your job.”

VERTICAL LINE TEST

SYMMETRIC function

About y-axis

​ -> even function $(x^2, x^4, x^4)$

w.r.t = with respect to

w.r.t origin / or (x=y)

​ ->odd function $(x^3, x^5, … x^{2n+1} )$

Domain of a function

Range of a function

Conposit function:

f(g(x) = f o g(x) -> f of g of x

x~y y~x (‘~’ use to relation)

g o f(x) = I(x) = x, and

f o g(y) = I(y) = y <– Identity function

f o $f^{-1} = f^{-1}$ o f = I

rate of change = $\frac{f(x_2-x_1)}{x_2-x_1} = \frac{change in\,f(x)\,w.r.t\,the\;change\;in\;x}{change\;in\;x}$

$ratio = \frac{a}{b}$

$rate = \frac{20RMB}{10kg\;apple} = \frac{2\;RMB}{1kg}$

截距 -> intercept

斜率 -> slope 割线 -> secant line

$y=x^3$	$y = x^2$	$y=x^n$
$y = x^{\frac{1}{3}}$	$y=x^{\frac{1}{2}}$	$y=x^{\frac{1}{n}}$
	four roots	n roots

in a triagle : “/” : Hypotenoccl?

​ “	” : opposite

​ “_”: adjacent

\[sin \theta = \frac{O}{H} SOH \\ C = \frac{A}{H} CAH \\ T = \frac{O}{A} TOA \\ sec \theta = \frac{1}{cos \theta} = \frac{H}{A} \\ csc \theta = \frac{1}{sin \theta} = \frac{H}{O} \\ cot \theta = \frac{1}{tan \theta} = \frac{A}{O} \\\]

radius

radial

radian 弧度

\[sin 2\theta = 2 sin \theta cos \theta \\ cos 2\theta = cos^2 \theta - sin^2 \theta \\ sec(\theta + 2\pi) = sec \theta \\ csc (\theta + 2\pi) = csc \theta \\ tan(\theta + \pi) = tan \theta \\ cot(\theta + \pi) = cot \theta \\ D(t) = 2.8 sin( \frac{2\pi}{365}(t-81)) + 12 \\\]

$\lim_{x \to a} f(x) = L \$ left sided: $lim_{\Delta x \to 0}f(a+\Delta x) = L \$ right side: $lim_{\Delta x \to 0}f(a+\Delta x) = M \$ If $L = M$ (both limits at $x=a$ coverage t)o same number)

then $lim_{x \to a} f(x) = L$

A function f is continuous a t “x=a”

1. f(a) exists
2. $lim_{x \to a}f(x)$ must exist
3. $lim_{x \to a}f(x)=f(a)$

right=sided continuous (Math definition)

​ = continuity (intuitive)

The Intermediate Value Theorem

f is continuous on the interval [a, b] and L is a number strictly between f(a) and f(b). Then therexists at least one number c in (a, b) satisfying f(c) = L.

Derivative of a function at a point

x = ‘a’ \(m_{tan} = \frac{dy}{dx} \vert_{x=a} = \lim_{h \to 0}\frac{f(a+h)-f(a)}{h} \\\) Ex. Use definition of derivative to find

\[\frac{df}{dx}\] \[(tan x)' = sec^2 x \\ (cot x)' = -csc^2 x \\ (sec x)' = sec x \cdot tan x \\ (csc x)' = -csc x \cdot cot x \\ \frac{d}{dx}(sin^{-1}x) = \frac{1}{\sqrt{1-x^2}} \\ \frac{d}{dx}(cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}} \\ \frac{d}{dx}(tan^{-1}x) = \frac{1}{1+x^2} \\ \frac{d}{dx}(cot^{-1}x) = -\frac{1}{1+x^2} \\ \frac{d}{dx}(sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}} \\ \frac{d}{dx}(csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2-1}} \\\]

$ X_c $ are critical point of a function $f’(x_c)$ = 0 or $f’(x_c)$ DNE.

if $f$ is continueous on $[a, b]$ And $f$ is differentiable on $(a, b)$

DEFINITION Critical Point:

An interior point c of the domain of f at which f’(c) = 0 or f’(c) fails to exist is called a critical point of f

FDT(First Derivative Test)

THEOREM 4.1 Extreme Value Theorem:

A function that is continuous on a closed interval [a, b] has an absolute maximum value and an absolute minimum value on that interval.

Absolute extreme values of a f in [a, b] Definition: If f{c} <= f(x) V x in [a, b] => f(c) is an absolute min value

If f’(x_c) = 0 or DNE(=>x_c is a critical point only finitely many) possible candidate for max/minnima (local) $+$ endpoint

Example of IVT usage: f(x)=x^5 + 5x+7 a. if f is continuous in [a, b] => x^5 + 5x + 7 is continuous on (-1, 0) b. f(a) < 0 < f(b) f(-1) = -3 < 0 < 5 = f(0) Then, IVT guarantees existence of a point $c \in (-1, 0) s.f. f(c) = 0$

1. Domian
2. x-intercept or y-intercepy
3. symmetric
4. slant, H.A, V.A.
5. f’ (intervals of increase, decrease) f” (concanity)

4.3 Graph of a function

\(y = \frac{2x^3+5x+5}{(x+1)^2}\) Chap 1. Domain: (-Inf, -1), (-1, Inf) Symmetic:f(-x) != f(x) != -f(x) Intercept x_intercept (y=0) y_intercept (x=0)

Chap 2.

• V.A: x=-1
• H.A. None
• S.A y = 2x

Mean Value Theorem Rolle’s Theorem

If

1. f(x) is continuous on [a, b]
2. f(x) is differentiable on (a, b)
3. f(a) = f(b) ?

Then there exists at least one c in (a, b) such that f’(c) = 0

f’(c) = 0 Mean height of f \(=> m_{tan} at "c" = m_{sec}(a, b)\)

Average rate of change aver (a, b) = instantaneous rate of change at some point at c in (a, b)