Up: [[OCW,OpenCourseWare,MOOC,MassiveOpenOnlineCourse]] [[미적분,calculus]] https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/index.htm [[TableOfContents]] = Lec 1 = Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007 https://www.youtube.com/watch?v=jbIQW0gkgxo 우선 [[이항정리,binomial_theorem]]에서 $(x+\Delta x)^n=x^n+nx^{n-1}\Delta x+\fbox{\mathrm{junk}}$ Junk는 $O\left((\Delta x)^2\right)$ , 작으므로 무시한다는 idea. $\frac{\Delta f}{\Delta x}=\frac1{\Delta x}\left((x+\Delta x)^n-x^n\right)$ $=\frac1{\Delta x}\left(\cancel{x^n}+nx^{n-1}\Delta x+O((\Delta x)^2)-\cancel{x^n}\right)$ $\longrightarrow^{\small\Delta x\to0}nx^{n-1}$ 따라서 $\fbox{\frac{d}{dx}x^n=nx^{n-1}}$ = Lec 2 = average change = $\frac{\Delta f}{\Delta x}$ instantaneous rate = $\frac{df}{dx}$ Examples 1. q = charge, dq/dt = current 2. s = distance, ds/dt = speed 3. T = temperature, dT/dx = temperature gradient 4. sensitivity of measurements f(x)가 x,,0,,에서 연속(continuous)이라 함은 $\lim{}_{x\to x_0}f(x)=f(x_0)$ Jump discontinuity : 좌극한과 우극한이 존재하나 다름. Removable discontinuity : 좌극한과 우극한이 같고 함수값만 동떨어져 있음. Infinite discontinuity Ex. 1/x [[쌍곡선,hyperbola]] : ±∞ // (이상 몇가지 [[불연속성,discontinuity]]) Proof of $\lim_{x\to x_0}f(x)-f(x_0)=0$ $=\lim\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)=f'(x_0)\cdot0=0 \rule{10}{10}$ = Lec 3 = https://www.youtube.com/watch?v=kCPVBl953eY $\frac{\rm d}{{\rm d}x}\sin x=\cos x$ Pf. $\frac{\sin(x+h)-\sin x}{h}=\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}$ $=\sin x\left(\frac{\cos h-1}{h}\right)+\cos x\left(\frac{\sin h}{h}\right)$ $h\to 0$ 이면 $=\sin x \cdot 0 + \cos x \cdot 1$ $=\cos x$ $\frac{\operatorname{d}}{\operatorname{d}x}\cos x=-\sin x$ Pf. $\frac{\cos(x+h)-\cos x}{h}=\frac{\cos x\cos h-\sin x\sin h-cos x}{h}$ $=\cos x\left(\frac{\cos h-1}{h}\right)-\sin x\left(\frac{\sin h}{h}\right)$ $h\to 0$ 이면 $=\cos x\cdot 0 -\sin x \cdot 1$ $=-\sin x$ = Lec 4 = https://www.youtube.com/watch?v=4sTKcvYMNxk Product rule 증명은 생략 $\frac{\text{d}}{\text{d}x}uv=\frac{du}{dx}{v}+u\frac{dv}{dx}$ Quotient rule 증명생략 $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$ Composition rule Chain rule [[연쇄법칙,chain_rule]] Higher derivative의 notation (u^^(n)^^, d^^n^^/dx^^n^^ u, D^^n^^u...) Ex. D^^n^^x^^n^^ = ? Dx^^n^^ = nx^^n-1^^ D^^2^^x^^n^^ = n (n-1) x^^n-2^^ D^^3^^x^^n^^ = n (n-1) (n-2) x^^n-3^^ ... D^^n-1^^x^^n^^ = n (n-1) … x^^1^^ So D^^n^^x^^n^^ = n! (constant) ∽ [[계승,factorial]] = Lec 5 = https://www.youtube.com/watch?v=5q_3FDOkVRQ Implicit differentiation $\frac{d}{dx}x^a=ax^{a-1}$ 지금까지 $a\in\mathbb{Z}$ 인 경우만 다루었으나 오늘은 $a\in\mathbb{Q}$ 인 경우를 다룰 예정. $\frac{d}{dx}y^n=\frac{d}{dx}x^m$ , ( $\frac{m}{n}=a$ 라고 가정 ) 양변을 미분하면 (좌변에 chain rule 적용된 것임) $\left(\frac{d}{dy}y^n\right)\frac{dy}{dx}=mx^{m-1}$ $ny^{n-1}\frac{dy}{dx}=mx^{m-1}$ $\frac{dy}{dx}=\frac{mx^{m-1}}{ny^{n-1}}$ $=\frac{m}{n}\frac{x^{m-1}}{(x^{m/n})^{n-1}}$ $=ax^{m-1-(n-1)\frac{m}{n}$ $=ax^{a-1}\quad\rule{10}{10}$ Example 2: $x^2+y^2=1$ $2x+2yy'=0$ $y'=-\frac{x}{y}$ Example 3: $y^4+xy^2-2=0$ explicit하게 하면 quadratic_formula를 쓰면 $y^2=\frac{-x\pm\sqrt{x^2-4(-2)}}{2}$ $y=\pm\sqrt{\frac{-x\pm\sqrt{x^2+8}}{2}}$ implicit하게 하면 $4y^3y'+y^2+x(2yy')-0=0$ $(4y^3+2xy)y'=-y^2$ $y'=\frac{-y^2}{4y^3+2xy}$ = Lec 6 = https://www.youtube.com/watch?v=9v25gg2qJYE $\frac{d}{dx}a^x= \frac{d}{dx}e^{x\ln a}=(\ln a)e^{x\ln a}=(\ln a)a^x$ $(\ln u)'=u'/u$ 양변에 로그를 취한 뒤 미분하는 것을 설명 ---- $\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$ $=e^{\left[\lim_{n\to\infty}\ln\left(\left(1+\frac{1}{n}\right)^{n}\right)\right]}$ $=e^1=e$ 근거는 다음과 같다. $\ln\left(\left(1+\frac1{n}\right)^n\right)=n\ln\left(1+\frac1{n}\right)$ 여기서 $\Delta x=\frac1n\to0$ 이라고 하면 $=\frac{1}{\Delta x}\ln(1+\Delta x)$ $=\frac{1}{\Delta x}\left(\ln(1+\Delta x)-\ln 1\right)$ $\Delta x\to 0$ 으로 가면 $=\left[\frac{d}{dx}\ln x\right]_{x=1}$ $=\left[1/x\right]_{x=1}$ $=1$ = Lec 7 = https://www.youtube.com/watch?v=eHJuAByQf5A $\frac{d}{dx}x^r=rx^{r-1}$ 증명 Method 1 $\frac{d}{dx}x^r$ $=\left(e^{r\ln x}\right)'$ $=e^{r\ln x}(r\ln x)'$ $=x^{r}\cdot\left(0\cdot\ln x+r\cdot\frac{1}{x}\right)$ $=x^{r}\cdot\left(\frac{r}{x}\right)$ $=rx^{r-1}$ Method 2 (log diff) $u=x^r$ $\ln u=r\ln x$ $\frac{u'}{u}=(\ln u)'=\frac{r}{x}$ $u'=u\cdot\frac{r}{x}=x^r\cdot\frac{r}{x}=rx^{r-1}$ = Lec 8 = (Lec 8 is exam 1) = Lec 9 = https://www.youtube.com/watch?v=BSAA0akmPEU Applications of differentiation Linear approximation ([[선형근사,linear_approximation]]) $\fbox{f(x)\approx f(x_0)+f'(x_0)(x-x_0)}$ i.e. Curve $y=f(x)$ $\approx y=f(x_0)+f'(x_0)(x-x_0)$ tangent line. Ex. f(x)=ln x, f'(x)=1/x $x_0=1,\;f(1)=\ln1=0,\;f'(1)=1$ $\ln x\approx 0+1\cdot(x-1)$ So, $\ln x\approx x-1$ (x가 1 근방일때) x,,0,,=0이면 $\fbox{f(x)\approx f(0)+f'(0)x$ 따라서 (x,,0,, ≈ 0)일 때는 sin x ≈ x cos x ≈ 1 exp x ≈ 1 + x ln(1 + x) ≈ x (1 + x)^^r^^ ≈ 1 + rx 이상 ≈의 좌변은 hard function, 우변은 easy function임을 볼 수 있음 Ex 2. ln(1.1) ≈ 1/10 ∵ ln(1+x) ≈ x, x=1/10 Ex 3. Find linear approx near x=0 (x≈0) of $\frac{e^{-3x}}{\sqrt{1+x}}$ $e^{-3x}(1+x)^{-1/2}$ $\approx(1-3x)\left(1-\frac12x\right)$ $=1-3x-\frac12x+\frac32x^2$ drop x^^2^^ terms (negligible) $\approx1-\frac72x$ Ex 4. (real life) 생략 = Lec 10 = Quadratic approximation (use these when linear is not enough) $f(x)\approx f(0)+f'(0)x+\frac{f''(0)}{2}x^2$ (when x≈0) x가 0에 가까울 때 다음 관계도 성립 $\sin x\approx x$ $\cos x\approx 1-\frac12x^2$ $\exp x\approx 1+x+\frac12 x^2$ $\ln(1+x)\approx x-\frac12 x^2$ $(1+x)^r\approx 1+rx+\frac{r(r-1)}{2}x^2$ = Lec 11 = https://www.youtube.com/watch?v=twzGBqPeW0M = Lec 12 = https://www.youtube.com/watch?v=YN7k_bXXggY = Lec 13 = https://www.youtube.com/watch?v=sRIDVAcoG5A tangent line: $y-y_0=m(x-x_0)$ x,,1,, is the x-intercept $0-y_0=m(x_1-x_0)$ $-\frac{y_0}{m}=x_1-x_0$ $x_1=x_0-\frac{y_0}{m}$ $x_1=x_0-\frac{f(x_0)}{f'(x_0)}$ 이것을 repeat(iterate) ([[뉴턴_방법,Newton_method]]) = Lec 14 = https://www.youtube.com/watch?v=4Q37iOyBq44 [[평균값정리,mean_value_theorem,MVT]] = Lec 15 = https://www.youtube.com/watch?v=-MI0b4h3rS0 [[적분표,integral_table]] = Lec 16 = https://www.youtube.com/watch?v=60VGKnYBpbg = Lec 17 = is test? = Lec 18 = https://www.youtube.com/watch?v=hjZhPczMkL4 적분 얘기 = Lec 19 = https://www.youtube.com/watch?v=1RLctDS2hUQ 치환 적분 = Lec 20 = https://www.youtube.com/watch?v=Pd2xP5zDsRw 적분을 사용한 [[평균,mean,average]] $\frac1{b-a}\int_a^bf(x)dx$ [[미적분학의기본정리,FTC]] = Lec 21 = https://www.youtube.com/watch?v=_JXPe2J069c [[로그,log]]의 적분을 쓴 정의 ... 다음 함수를 오래 설명 $F(x)=\int_0^x e^{-t^2}dt$ 그래프: arctan 유사한 모양, horizontal asymptote는 $\lim_{x\to\infty}F(x)=\frac{\sqrt{\pi}}{2},\; \lim_{x\to-\infty}F(x)=-\frac{\sqrt{\pi}}{2}$ $F'(x)=e^{-x^2}$ 그래프 : 정규분포 모양 $F''(x)=-2xe^{-x^2}$ $\begin{cases}<0,x>0\\>0,x<0\end{cases}$ 이며 기함수 즉 F(-x)=-F(x) 여기서 [[오차함수,error_function]] (curr. goto [[함수,function#s-17]]) 이 나옴 $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt=\frac{2}{\sqrt{\pi}}F(x)$ 그리고 다음을 Fresnel integral 이라 함 $C(x)=\int_0^x\cos(t^2)dt$ $S(x)=\int_0^x\sin(t^2)dt$ $Li(x)=\int_2^x\frac{dt}{\ln t}$ Li(x)는 (x보다 작은 [[소수,prime_number]])에 비례 ... 곡선 사이의 면적을 구하는 것 설명 = Lec 22 = https://www.youtube.com/watch?v=ShGBRUx2ub8 얇게 잘라 부피를 구하는 것을 설명 (volumes by slicing) $\Delta V\approx A\Delta x$ (one slice) $dV=A(x)dx$ $V=\int A(x)dx$ 회전해서 생긴 부피 설명 (solids of revolution) = Lec 23 = is test? = Lec 24 = https://www.youtube.com/watch?v=jBkXbAgMj6s = Lec 25 = https://www.youtube.com/watch?v=zUEuKrxgHws = Lec 26 = is an exam session = Lec 27 = https://www.youtube.com/watch?v=Bv9kVDcj7yo = Lec 28 = https://www.youtube.com/watch?v=CXKoCMVqM9s = Lec 29 = https://www.youtube.com/watch?v=HgEqXhsIq_g rational function = [[다항식,polynomial]] / 다항식 [[부분분수,partial_fraction]] 부분분수분해,partial fraction decomposition 부분분수전개,partial fraction expansion = Lec 30 = https://www.youtube.com/watch?v=aeXp1zC6Hls $\int uv'dx=uv-\int u'vdx$ $\int_a^b uv'dx=uv|_a^b-\int_a^b u'vdx$ Ex. $\int\ln x dx=?$ Let $u=\ln x,\; u'=\frac1x;\; v=x,\, v'=1$ $=\int uv'dx=uv-\int u'v dx$ $=\int\ln x dx=x\ln x-\int \frac1x x dx$ $=x\ln x-x+C$ = Lec 31 = https://www.youtube.com/watch?v=TpWQlKHPyJ4 arc_length 피타고라스 정리에서 $ds^2=dx^2+dy^2$ $ds=\sqrt{dx^2+dy^2}$ dx를 밖으로 끄집어내면 $ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$ 그래서 [[곡선,curve]] 길이 공식이 이 모양인 것이다. arclength는 ([[호길이,arclength]]) $\int ds$ $=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$ $=\int_a^b\sqrt{1+f'(x)^2}dx$ 표면적,surface_area AKA 겉넓이 회전체의 겉넓이 구의 겉넓이 radius=a일 때 $y=\sqrt{a^2-x^2}$ $y'=\frac{-x}{\sqrt{a^2-x^2}}$ $1+\frac{x^2}{a^2-x^2}=1+(y')^2$ $=\frac{a^2-x^2+x^2}{a^2-x^2}=\frac{a^2}{a^2-x^2}$ 따라서 area = $\int_{x_1}^{x_2}2\pi y ds$ $=\int_{x_1}^{x_2}2\pi\sqrt{a^2-x^2}\sqrt{\frac{a^2}{a^2-x^2}}dx$ $=\int_{x_1}^{x_2}2\pi a dx$ $=2\pi a(x_2-x_1)$ 따라서 $4\pi a^2$ 유추 가능 parametric_curve $\begin{cases}x=x(t)\\y=y(t)\end{cases}$ = Lec 32 = https://www.youtube.com/watch?v=XRkgBWbWvg4 [[극좌표,polar_coordinate]] = Lec 33 = https://www.youtube.com/watch?v=BGE3wb7H2PA Exam 4 review = Lec 34 = exam? = Lec 35 = https://www.youtube.com/watch?v=PNTnmH6jsRI 극한이 [[부정형,indeterminate_form]]일 때 활용하는 [[로피탈_정리,L_Hopital_s_rule]] Version 1 $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$ provided $f(a)=g(a)=0$ and the right-hand limit exists. = Lec 36 = https://www.youtube.com/watch?v=KhwQKE_tld0 [[이상적분,improper_integral]] Def: $\int_a^\infty f(x)dx=\lim_{N\to\infty}\int_a^Nf(x)dx$ The integral converges if limit exists; diverges if not. Example: $\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$ $\int_1^{\infty}\frac{dx}{x^p}$ diverges if $p\le 1$ $\int_1^{\infty}\frac{dx}{x^p}$ converges if $p>1\quad \left(=\frac1{p-1}\right)$ 이건 판정법 내용인가? Limit comparison If $f(x)\sim g(x)$ (similar) as $x\to\infty$ (f~g as x→∞ means f(x)/g(x)→1) then $\int_a^{\infty}f(x)dx\;\textrm{and}\;\int_a^{\infty}g(x)dx$ either both converge or both diverge. Ex. $\int_0^{\infty}\frac{dx}{\sqrt{x^2+10}}$ $\sqrt{x^2+10}\sim\sqrt{x^2}=x$ 이므로 $\sim\int_1^{\infty}\frac{dx}{x}$ - diverges. Ex. $\int_{10}^{\infty}\frac{dx}{\sqrt{x^3+3}}$ $\frac1{\sqrt{x^3+3}}\sim\frac1{\sqrt{x^3}}=\frac1{x^{3/2}}$ $\int_{10}^{\infty}\frac{dx}{x^{3/2}}$ - convergent. = Lec 37 = https://www.youtube.com/watch?v=MK_0QHbUnIA Improper integrals (2nd kind) Ex 1 $\int_0^1\frac{dx}{\sqrt{x}}=\int_0^1x^{-\frac12}dx=\left[2x^{\frac12}\right]_0^1=2-0$ convergent Ex 2 $\int_0^1\frac{dx}{x}=\left[\ln x\right]_0^1=\ln1-\ln0^+=0-(-\infty)=+\infty$ divergent In general, $\int_0^1\frac{dx}{x^p}=\frac1{1-p}$ if $p<1$ diverges if $p\ge 1$ Contrast $\frac1{x^{1/2}}\ll \frac1x \ll \frac1{x^2} \quad \textrm{as}\; x\to\0^{+}$ $\frac1{x^{1/2}}\gg \frac1x \gg \frac1{x^2} \quad \textrm{as}\; x\to\infty$ [[급수,series]] [[무한급수,infinite_series]] [[기하급수,geometric_series]] $1+\frac12+\frac14+\frac18+\cdots=2$ $1+a+a^2+a^3+\cdots=\frac1{1-a}\;(-1 \frac12 + \frac13 + \cdots + \frac1N = S_N - 1$ $\ln N < S_N < (\ln N)+1$ ---- Integral Comparison If $f(x)$ is decreasing, $f(x)>0,$ then $\left| \sum_{n=1}^{\infty} f(n) - \int_1^{\infty} f(x)dx \right| < f(1)$ and $\sum_{n=1}^{\infty}f(n)$ and $\int_1^{\infty}f(x)dx$ converge or diverge together. (rel. [[적분판정법,integral_test]]) ---- Limit Comparison If $f(n)\sim g(n) \left(\text{ (i.e. } \frac{f(n)}{g(n)}\to 1\text{ as } n\to\infty \right)$ and $g(n)>0 \; \forall n $ then, $\sum f(n),\,\sum g(n)$ either both converge or both diverge. Ex. $\sum_{n=0}^{\infty}\frac1{\sqrt{n^2+1}} \;\leftrightarrow\; \sum\frac1{\sqrt{n^2}} = \sum\frac1n $ diverges. Ex. $\sum_{n=2}^{\infty}\frac1{\sqrt{n^5-n^2}} \;\leftrightarrow\; \sum\frac1{\sqrt{n^5}} = \sum\frac1{n^{5/2}}$ converges. (rel. [[극한비교판정법,limit_comparison_test]]) = Lec 38 = https://www.youtube.com/watch?v=wOHrNt9ScYs (김홍종 미적분학 1+에도 언급된 '블럭을 쌓아 한강을 건널 수 있는가?' 그 문제) 도중에 [[greedy_algorithm]] 언급 결론은 가능하다는 것. 다만 앞의 테이블을 건너려면 달의 거리의 두 배만큼의 높이를 쌓아야 함. ---- Power Series ([[멱급수,power_series]]) $1+x+x^2+x^3+\cdots=\frac1{1-x}\;\;|x|<1$ ([[기하급수,geometric_series]]) $1+x+x^2+x^3+\cdots=S$ 라 하자 양변에 $x$ 를 곱하면 $x+x^2+x^3+\cdots=Sx$ 위 두 식에 대해, 위 식에서 아래 식을 빼면 $1=S-Sx$ $1=S(1-x)$ $\frac1{1-x}=S$ Reasoning (is) incomplete because it requires $S$ exist. e.g. $x=1,$ $1+1+1+\cdots=S$ $1+1+1+\cdots=S\cdot 1$ $\infty-\infty=\infty-\infty$ ---- General Power Series ([[멱급수,power_series]]) $a_0+a_1x+a_2x^2+a_3x^3+\cdots=\sum_{n=0}^{\infty}a_n x^n$ $|x|R,\;\sum a_n x^n$ diverges $|x|=R:$ very delicate borderline - not used by us 그래서 우린 수렴반지름 안에서만 머무를 것이다. $|a_n x^n|\to 0$ exponentially fast for $|x|R$ (R 밖에 있을 때는 0으로 가지도 않는다) 이건 모든 멱급수(power series)에 적용된다. ---- Rules for convergent power series are just like polynomials. $f(x)+g(x),\;f(x)g(x),\;f(g(x)),\;\frac{f(x)}{g(x)},\;\frac{d}{dx}f(x),\;\int f(x)dx$ $\frac{d}{dx}\left(a_0+a_1x+a_2x^2+a_3x^3+\cdots\right)=a_1+2a_2x+3a_3x^2+\cdots$ $\int\left(a_0+a_1x+a_2x^2+\cdots\right)dx=C+a_0x+a_1x^2/2+a_2x^3/3+\cdots$ Taylor's formula $f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$ $\sum_{n=0}^{\infty}\fbox{\frac{f^{(n)}(0)}{n!}}x^n$ 사각형 부분이 $a_n$ $f(x)=a_0+a_1x+a_2x^2+a_3x^3+\cdots$ $f'(x)=a_1+2a_2x+3a_3x^2+\cdots$ $f^{(2)}(x)=2a_2+3\cdot 2 a_3x+\cdots$ $f^{(3)}(x)=3\cdot2a_3+4\cdot3\cdot2a_4x+\cdots$ 여기서 $x=0$ 을 넣으면, $f^{(3)}(0)=3\cdot 2 a_3$ $\frac{f^{(3)}(0)}{3!}=a_3$ In general, $a_n=\frac{f^{(n)}(0)}{n!}$ [[지수함수,exponential_function]]를 예로 들면 $f(x)=e^x,\;f'(x)=e^x,\;f''(x)=e^x,\;\ldots$ $f^{(n)}(x)=e^x$ $f^{(n)}(0)=\left.e^x\right|_{x=0}=1$ 그래서 분자는 모두 1이다. $e^x=\sum_{\small n=0}^{\infty}\frac1{n!}x^n$ $e=e^1=1+1+\frac1{2!}+\frac1{3!}+\frac1{4!}+\cdots$ $\sin x \approx x$ $\cos x \approx 1-\frac{x^2}{2}$ $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$ = Lec 39 = https://www.youtube.com/watch?v=--lPz7VFnKI [[테일러_급수,Taylor_series]] Ex 1. $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$ Ex 2. $\frac1{1+x}=1-x+x^2-x^3+\cdots$ Ex 3. $\sin'x=\cos x$ $\sin{}'{}'x=-\sin x$ $\sin^{(3)}x=-\cos x$ $\sin^{(4)}x=\sin x$ $\sin(x)=x-\frac1{3!}x^3+\frac1{5!}x^5-\cdots$