Ver.2.2 20.07.2

3 200 6 2

4 ) 2) 3) 4) 5) (S45 9 ) ( 4) III 6) 7) 8) 9) ) 2) 3) 4) BASIC 5) 6) 7) 8) 9)

..2 3.2. 3.2.2 4.2.3 5.2.4 6.3 8.3. 8.3.2 8.3.3 9.4 2.5 3.6 5 2.6. 5.6.2 6.6.3 9.6.4 20.6.5 2.6.6 23.6.7 RC 24 29 2. 29 2.2 29 2.3 30 2.4 30 2.5 30 2.6 3 2.7 33

ii 3 35 3. 35 3.2 Java 35 A 4 A 4 B C D E F G H I J K 4 4 42 43 44 46 48 50 50 ɛ δ 5 B 53 A 53 B C 55 56 C. 56 C.2 57 C.3 59 C.4 60 C.5 62 C.6 63 C.7 64 C.8 65 C.9 65 C 67 A 67 B C 68 69 C. 69 C.2 7 C.3 7 C.4 72 C.5 73

iii D 75 E 77 79

..2 23 2. 32 3. 39

. 0.2 step6 4.3 8.4 9.5 27 3. ( ) 38 A. 42 A.2 43 A.3 44 A.4 45 A.5 e 48 B. 53 B.2 55 B.3 e 62 C. 68

. x y = f(x) y y, y, y (n) n F (x, y, y, y,, y n ) = 0 (.) dy dt = ay y ay = 0 ay + by + cx = 0 (ay ) 2 = 0! d 2 2 y dx 2 = 0 x t x y

2 y = ax y = a y = ax a a = y y = y x y = x 2 xy 2y = 0 y = 2x y y x 2x 2 x 2 = 2x 2 2x 2 = 0 y = e ax y = ay y = ae ax y y ae ax = a e ax

.2 3.2.2. x dy = f(x) (.2) dx dy dx = f(x)dx dx dy dx dx = f(x)dx + C dy = f(x)dx + C y = f(x)dx + C y = f(x)dx + C (.3) (x, y) = (, 5) y = 2x + y = (2x + )dx + C y = x 2 + x + C C C y = x 2 + x + C 5 = 2 + + C C = 5 2 = 3 y = x 2 + x + 3

4.2.2 x y dy = f(x) g(y) (.4) dx dy dx = f(x)g(y)dx dx dy dx = f(x)dx g(y) dx dy g(y) dx dx = f(x)dx + C g(y) dy = f(x)dx + C g(y) dy = f(x)dx + C (.5) 2 (x, y) = (, ) y = x y ydy = xdx + C 2 y2 = 2 x2 + C x 2 + y 2 = 2C = C C C x 2 + y 2 = C 2 + 2 = C C = + = 2 x 2 + y 2 = 2

.2 5.2.3 «dy y dx = f y x x = u dy «y dx = f x y = xu y u 2 y = f(u) y = xu y = (x) u + x(u) y = u + x u 2 u + x u = f(u) x u = f(u) u u = f(u) u x f(u) u u = x x 0 x R f(u) u du dx dx = R x dx + C (f(u) u) 0 f(u) u du = log e x + C (.6) u = y x 3 x dy dx = x + y dy dx = + y y (x 0) x x = u dy dx = + u f(u) = u + R (.6) f(u) = u + (u + ) u du = log e x + C R du = log e x + C u = log e x + C u = y x y x = log e x + C y = x(log e x + C)

6.2.4 ( y ( dy ) y dx ) dy + p(x)y = q(x) (.7) dx q(x) 0 dy + p(x)y = 0 (.8) dx dy y = p(x)dx log e y = p(x)dx + C y = ce R p(x)dx (.9) c ( ) c x C(x) y = C(x)e R p(x)dx (.0) (.7) {C(x)e R p(x)dx } + p(x){c(x)e R p(x)dx } = q(x) C (x)e R p(x)dx + C(x){e R p(x)dx } + p(x)c(x)e R p(x)dx = q(x) C (x)e R p(x)dx + C(x)( p(x))e R p(x)dx + p(x)c(x)e R p(x)dx = q(x) C (x)e R p(x)dx = q(x) C (x) = q(x)e R p(x)dx

.2 7 C(x) = q(x)er p(x)dx dx + C (.0) q(x) 0 y = e R j p(x)dx ff p(x)dx q(x)er dx + C (.) 4 y y = x (.7) p(x) =, q(x) = x (.) y = e R j ( )dx R j y = e dx ff ( )dx xer dx + C ff xe R dx dx + C j ff y = e x xe x dx + C C.28 j ff y = e x xe x dx + C ff y = e j xe x x ( e x )dx + C ff y = e j xe x x + e x dx + C y = e x ( xe x e x + C) y = xe x x e x x + Ce x y = x + Ce x

8.3.3. (x, y) y (0, 2) dy y dx.3.2 dy dx = y (.2) dy y dx = x dy y dx dx = dx y dy = dx y log e y = x + C (C ) y = ±e x+c = ±e C e x = α e x ( α = ± e C ) (.3) α α (0, 2) (0, 2) x = 0 y = 2 (.3) 2 = α e 0 α = 2 (x, y) y (0, 2) y = 2 e x (.4)

.3.3.3 9 (.4) (.2) dy dx = y dy = y dx x 0 = 0.0, y 0 = 2.0 dx = 0. x = 0., (= x 0 + dx = 0.0 + 0.), y = 2.2, (= y 0 + y 0 dx = 2.0 + (2.0 0.)) x 2 = 0.2, (= x + dx = 0. + 0.), y 2 = 2.42, (= y + y dx = 2.2 + (2.2 0.)) x 3 = 0.3, (= x 2 + dx = 0.2 + 0.), y 3 = 2.662, (= y 2 + y 2 dx = 2.42 + 2.42 0.)). dx 0. 0.0 dx dy i. yr.data y.data dx 0. y2.data dx 0.0

0 45 40 35 30 25 20 5 0 5 0 y.data yr.data y2.data 0 0.5.5 2 2.5 3.

.3. x(i) dy(i+) y(i+) y(i+) dx(=0.0) =y(i) dx(=0.) =y(i)+dy(i+) 2 e x 0.0 2.0 2.0 2.0 0. 0.2 2.2 2.2092442508 2.20342 0.2 0.22 2.42 2.4403800799 2.442806 0.3 0.242 2.662 2.6956978307 2.69978 0.4 0.2662 2.9282 2.9777274672 2.983649 0.5 0.29282 3.2202 3.2892636437 3.297443 0.6 0.32202 3.54322 3.633393397 3.644238 0.7 0.35432 3.897434 4.035267368 4.027505 0.8 0.389743 4.28778 4.4334304344 4.45082 0.9 0.42878 4.75895 4.8972653493 4.99206.0 0.47590 5.87485 5.4096276588 5.436564 3.0 3.7269 34.898805 39.5769325238 40.7074

2.4 (.5) f(x, y) dy = f(x, y) (.5) dx (x, y) D (x 0, y 0 ) D dy dx = f(x, y), y 0 = y(x 0 ) (.6) (.6) x 0 (.5) f(x, y) (x, y) D y f(x, y 2 ) f(x, y ) < L y 2 y (L > 0) (.7) (x 0, y 0 ) D x 0 (.6) y(x)

.5 3.5 step: step2: t = 0 2g dy t y(t) dt dy dt t dy dt = ky (k ) y t k (k < 0) step3: 88Ra 2 26 k.4x0 [s ] dy dt = ky (k ) dy y dt = k dy y dt dt = kdt y dy = kdt log e y = kt + C y = ±e kt+c y = ±e C e kt y = c e kt step4: t = 0 2g

4 y(0) = 2 ce k 0 = 2 ce 0 = 2 c = 2 y(t) = 2e kt step5: step6: y (t) = dy dt = 2e kt = 2ke kt = k 2e kt = ky y(0) = 2e k 0 = 2e 0 = 2 dy = ky y(0) = 2 dt y c, t > 0 y = c e t c y = c e 0 0.2 y = c e t step6 t step7: step8:.2 step3 k < 0 c e t k =

.6 5.6.6. 8.0 2.0.5 a a + b 8.0 : 2.0 = a + b : b 2.0 (a + b) = 8.0 b b = 3 a.5 a t da dt =.5 (.8) t db dt = 3 da dt (.8) db dt = 3.5.5 = 0.5 3 /

6.6.2 (I) 798 step: step:2 step:3 N = N(t) t t αn t βn t t N N = αn t βn t = (α β)n t = γn t step:4 step:5 dn dt = γn dn N dt = γ step:6 dt dn N dt dt = γdt N dn = γdt log e N = γt + C N = e γt+c N = e C e γt t = 0 N = N 0 N 0 = e C e γ 0 N 0 = e C N = N 0 e γt

.6 7 (II) 837 step: step:2 step:3 step:4 step:5 N n N N N n dn dt dn dt = γn( N N n ) step:6 dn dt = γn( N ) N n dn N dt = γ( N ) N n dn N N dt = γ N n dn N N dt dt = γdt N n N N dn = γdt N n N + N n N dn = N n γdt log e N log e N N n «= γt + C log e N N Nn «= γt + C N N «= e γt+c = e C e γt = ce γt N n

8 step:7 t = 0 N = N 0 N 0 N 0 N n «= ce γ0 = ce 0 = c c N N Nn «= N 0 N 0 N n «e γt N N N «= 0 «e γt Nn N Nn N 0 N n N n N N n N = N 0 e γt N n N 0 N n N = N n N 0 N N 0 e j ff γt N = Nn N 0 e γt N n N n N N = N n = + j Nn N 0 j Nn + N 0 ff ff e γt e γt step:8 N = N n j ff (.8) Nn + e N γt 0.3 N n N N = N n o +n Nn e γt N 0 N 0 0.3 t

.6 9.6.3 t x x v = lim t 0 t (.9) = dx dt (.20) v v a = lim t 0 t = dv dt (.2) a = dv dt = d j ff dx dt dt = d2 x dt 2 (.22) 2.4.4

20.6.4 a = g g dv dt = g (.23) dv = g dt dv = g dt v = g t + C t = 0 v = v 0 v 0 = g 0 + C C = v 0 v = g t + v 0 (.24) v = dx dt = g t + v 0 dx dt = g t + v 0 dx = g tdt + v 0 dt x = 2 g t2 + v 0 t + C t = 0 x = 0 t = 0 C = 0 x = 2 g t2 + v 0 t (.25)

.6.5.6 2 S s t = 0 y(0) = h ( ) ) v Torricelli v = k 2gy g = 9.8m/s 2 k dt vdt s svdt = sk 2gydt dy Sdy = sk p 2gydt S sk S sk S sk 2g y dy = dt 2g dy = y dt 2g 2 ydy = t + C S sk s 2 y = t + C g t = 0, y(0) = h C S s 2 h = 0 + C sk g C = S s 2 h sk g

22 S s 2 S y = t sk g sk s S 2 S y = sk g sk r sk g y = S 2 S sk s 2 h g s 2 h t g s! 2 h t g r sk g y = h S 2 t r sk g S 2 t = h y t = ( h y) S sk s 2 g

.6 23.6.6 E R, E L, E C V R Ω I A L H t s C F Q C. R, E R E R = R I (.26) 2. L, E L E L = L di (.27) dt 3. C, E C E C = Q C I = dq dt (.28) (.29).2 m L 2 v I 3 x Q 4 F = m dv dt E = LdI dt 5 2 mv2 2 LI2

24.6.7 RC E R C 0 t Q 0 Q = 0. a. t (.29) I = dq dt b. (.26,.29) RI = R dq dt c. (.28) Q C d. 2. R dq dt + Q C = E

R dq dt = E Q C dq dt = E Q «R C dq dt = (EC Q) RC dq dt = (Q CE) RC dq Q CE dt = (Q CE 0) RC dq Q CE dt dt = dt RC Q CE dq = dt RC (C.29) log e Q CE = RC t + C Q CE = e RC t+c (Q CE < 0) Q + CE = e C e RC t Q + CE = A e RC t Q = CE A e RC t.6 25 t = 0, Q = 0 0 = CE A e RC 0 A = CE Q = CE CE e t RC Q = CE e t RC (Q CE > 0) Q CE = e C e RC t Q CE = A e RC t Q = CE + A e RC t t = 0, Q = 0

26 0 = CE + A e RC 0 A = CE Q = CE CE e t RC Q = CE e t RC (Q CE 0) Q = CE e t RC (.30) I.29 I = dq n dt = CE e t o CE RC = ( RC e t RC ) = E R e t RC t I 3. E = 0 R dq dt + Q C = 0 Q = B e RC t (.3) t 0 Q Q = CE e t 0 RC t 0 = Q ; CE e RC ; CE e ; CE ( 0) = CE (t = t 0 ) Q = B e RC t CE = B e RC t 0 B = CE e RC t 0 B Q = CE e t 0 RC e t RC Q = CE e t 0 t RC Q = CE e (t t 0 ) RC

.6 27 I.29 I = dq dt = j ff CE e (t t 0 ) RC = CE 4..5 CE = 8 Q RC e (t t0) RC = E R e (t t 0 ) RC Q = CE e t RC Q = CE e t t 0 RC 0 t 0 = 0 E = 4, R =, C = 2, t 0 = 0 t 0 :.5 t

2 2. f(t) F (s) F (s) f(t) F (s) = 0 f(t)e st dt (t 0) (2.) s F (s) = L[f(t)] (2.2) 2.2. f(t) = L() = 0 e st dt =» s e st 0 = s (s > 0) (2.3) 2. f(t) = t [ (C.28) ] L(t) = 0» t e st dt = 3. f(t) = e at + st s 2 e st 0 = s 2 (s > 0) (2.4)

30 2 L(e at ) = 0 e at e st dt = 0 e (a s)t dt = " e (a s)t a s # 0 = s a (s > a) (2.5) 2.3. 2. tf(t) 3. f(t)/t 4. f (t) 5. f (t) L[af(t) + bg(t)] = al[f(t)] + bl[g(t)] (2.6) L[f(t)] = L[F (s)] (2.7) L[tf(t)] = L[F (s)] (2.8)» f(t) s L = F (s)ds (2.9) t L[f (t)] = sf (s) f(0) (2.0) L[f (t)] = s 2 F (s) sf(0) f (0) (2.) 2.4 L [F (s)] = f(t) (2.2) f(t) 2.2 2.5 (.5 ). f(t)

dy dt = ky (k, y = f(t), f(0) = 2) y = ky 2.6 3 2. F (s) L[y ] = L[ky] 3. F (s) L[y ] = kl[y] sf (s) f(0) = kl[y] sf (s) f(0) = kf (s) (2.7) sf (s) 2 = kf (s) sf (s) 2 kf (s) = 0 (s k)f (s) = 2 F (s) = 2 (s k) (2.6 ) (2.0 ) ( f(0) = 2) 4. f(t) f(t) = 2 e kt (2.5 ) 2.6 2.2 L[f(t)] f(t)

32 2 2. 2 t f(t) L[f(t)] s s 3 e at s a 4 cos ωt 5 sin ωt 6 e αt cos βt 7 e αt sin βt 8 f(t) = 0, (0 t < a) f(t) =, (t a) s > 0 s 2 s > 0 s > a s s 2 + ω 2 s > 0 ω s 2 + ω 2 s > 0 s α (s α) 2 + β 2 s > α β (s α) 2 + β 2 s > α s e as s > 0 ) ) Mathematics Handbook for Science and Engineering ( Råde,Westergren Springer)

2.7 33 2.7.6.7 E R C 0 t Q 0 Q = 0 R dq dt + Q C = E» L R dq dt + Q = L[E] C y dq dt y Q» L R Q + Q = L[E] C RL[Q ] + L[Q] = EL[] C F (s) = E s C CRs + = E s R{sF (s) f(0)} + C F (s) = E s RsF (s) + C F (s) = E (f(0) = 0) s CRs + F (s) = E C s F (s) = E R 0 C s + CR = E B @ R s C s + A = E B @ R s C s + A» CR CR RC CR (s + α)(s + β) = β α s + α «s + β 0 0 CR 0 B @ s + 0 s + CR C A = E R CR 0 B @ s s + CR C A

34 2 2 L [F (s)] = L 6 4 E R CR 0 B @ s s + CR 3 C7 A5 = ECL 6 4 s 7 s + 5 0 2 CR 3» = EC B @ L L 6 s 4 s «7C 5A CR 2 3 f(t) = Q = EC e t CR

3 Java gnuplot 3. y 0 = n y (3.) x 0 = n x (n x, n y ) (3.2) k = h f(x i, y i ) (3.3) k 2 = h f(x i + h 2, y i + k 2 ) (3.4) k 3 = h f(x i + h 2, y i + k 2 2 ) (3.5) k 4 = h f(x i + h, y i + k3) (3.6) y i+ = y i + 6 (k + 2k 2 + 2k 3 + k 4 ) (3.7) x i+ = x i + h (3.8) x i y i Java Java C 3.2 Java II

36 3 Java for 3. dn C dt = γn( N ) N n func II func II (.8) runge.data runger.data RungeRun.bat javac Runge.java pause OK java Runge pause OK rem ()gnuplot>set terminal postscript eps rem (2)gnuplot>set output "runge.eps" rem (3)gnuplot>plot..\..\tpc-math\runge\runge.data with lines rem (4)gnuplot>replot..\..\tpc-math\runge\rungeR.data with lines rem GNUPLOT..\..\gnuplot\bin\wgnuplot.exe. Java Javac 2. JavaVM[Java java ] runge.data 3. GNUPLOT 3 pause pause rem eps TeX 3.?? gnuplot eps gnuplot Windows

Runge.java 3.2 Java 37 import java.io.*; class Runge { private static double t,x,y,y_next,h,tmax,n0,nmax,g; private static double k,k2,k3,k4; public static void main(string [] args) { String outfilename = "runge.data"; // String outfilenamer = "runger.data";// x=0.0; // x N0=3.0; // y y=n0; // h=0.00; // dt...dx tmax=30.0; //x g=0.5; //g...gamma Nmax=00.0; //y Runge runge=new Runge(); // try { FileWriter fw = new FileWriter(outFileName); // fw.write( x + " " + y + "\n" ); //(x(0),y(0)) FileWriter fwr = new FileWriter(outFileNameR); // fwr.write(x+" "+runge.funcriron(x,n0)+"\n"); // (x(0),y(0)) for(t = 0; t < tmax; t += h) { k = h * runge.func(x, y); k2 = h * runge.func(x + (h / 2.0), y + k / 2.0); k3 = h * runge.func(x + (h / 2.0), y + k2 / 2.0); k4 = h * runge.func(x + h, y + k3); y += (k + 2.0 * k2 + 2.0 * k3 + k4) / 6.0; //y(i+)=y(i)+dy x += h; // x(i+)=x(i)+dx fw.write ( x + " " + y + "\n" ); // (x(i),y(i)) fwr.write(x+" "+runge.funcriron(x,n0)+"\n");// (x(i),y(i)) } fw.close(); fwr.close(); } catch(exception e) { System.out.println(" "); } } } double func(double x,double N) { // return g * N * (.0 - N / Nmax); } double funcriron(double x,double N0) { // return Nmax / (+((Nmax/N0)-)*Math.exp(-g*x)) ; }

38 3 00 N..\..\tpc-math\runge\runge.data 90 80 70 60 50 40 30 20 0 0 0 5 0 5 20 25 30 35 runge.data 3. ( ) 0.0 3.0 0.000 3.00455349750447 0.0020 3.0029368003934 0.0030 3.0043680786764524 0.0040 3.005825474003739 0.0050 3.0072835543828824 4.998000000000004 27.34678468265506 4.999000000000004 27.3567990255354 5.000000000000004 27.366657436770502 9.997999999999898 82.0965360742 9.998999999999898 82.038794850906 9.999999999999897 82.2250046675 0.000999999999896 82.85687746497 29.9980000000368 99.99900993605378 29.9990000000368 99.99900430957 30.0000000000368 99.999009256306 t

3.2 Java 39 3. t N 0.0 3.0 3.0 2 0.000 3.00455349750447 3.0045534975044 3 0.0020 3.0029368003934 3.002936800393 4 0.0030 3.0043680786764524 3.004368078676452 5 0.0040 3.005825474003739 3.0058254740037387 6 0.0050 3.007283554382882 3.0072835543828824 7 0.0060 3.00874232046226 3.0087423204622 8 0.0070 3.0020774998 3.0020774998 9 0.0080 3.066908839405 3.066908839405 0 0.00900000000000000 3.03227324344954 3.03227324344954 4999 4.998000000000004 27.34678468265506 27.34678468265623 5000 4.999000000000004 27.3567990255354 27.3567990255366 500 5.000000000000004 27.366657436770502 27.366657436770623 9999 9.99799999999989 82.0965360742 82.09653607364 0000 9.998999999999898 82.038794850906 82.03879485085 000 9.999999999999897 82.2250046675 82.225004669 0002 0.000999999999896 82.85687746497 82.85687746438 29999 29.9980000000368 99.99900993605378 99.99900993605387 30000 29.9990000000368 99.99900430957 99.999004309572 3000 30.0000000000368 99.999009256306 99.99900925636

A A B a 2 + 2ab + b 2 = (a + b) 2 (A.) a 2 2ab + b 2 = (a b) 2 (A.2) a 2 b 2 = (a + b)(a b) (A.3) x 2 + (a + b)x + ab = (x + a)(x + b) (A.4) acx 2 + (ad + bc)x + bd = (ax + b)(cx + d) (A.5) a 3 + b 3 = (a + b)(a 2 ab + b 2 ) (A.6) a 3 b 3 = (a b)(a 2 + ab + b 2 ) (A.7) a 3 + 3a 2 b + 3ab 2 + b 3 = (a + b) 3 (A.8) a 3 3a 2 b + 3ab 2 b 3 = (a b) 3 (A.9) a n b n = (a b)(a n + a n 2 b + a n 3 b 2 + +a 2 b n 3 + ab n 2 + b n ) (A.0) ax 2 + bx + c = 0 C x = b ± b 2 4ac 2a (b 2 4ac 0) (A.) f( x) = f(x) (A.2) f( x) = f(x) (A.3)

42 A y = x 2, cos θ y = x, sin θ D ([0, 0] > [a, b]) y = (x a) 2 + b (A.4) y b = (x a) 2 (A.5) y y = x 2 y = (x a) 2 + b b 0 a x A.

E 43 E y = x 2 x = y x y y = x y = x 2 y = x y = x 45 y = x 2 y = x y 0 y y = x 2 y = x y = x 0 x A.2 y = f(x) x = g(y)

44 A F y y r P (x, y) r r θ 0 x r x A.3 r 360 = θ rad. sinθ = y r (A.6) cosθ = x r (A.7) tanθ = y x (A.8)

F 45 y y = tanθ y = sinθ π 2 0 π 3π 2 π 2 2π x - y = cosθ

46 A G tanθ = sinθ cosθ sin 2 θ + cos 2 θ = + tan 2 θ = cos 2 θ sin( θ) = sinθ cos( θ) = cosθ tan( θ) = tan θ (A.9) (A.20) (A.2) (A.22) (A.23) (A.24) sin( π 2 θ) = cosθ (A.25) cos( π θ) = sinθ 2 (A.26) tan ( π 2 θ) = tan θ sin (π ± θ) = sin θ cos (π ± θ) = cos θ tan (π ± θ) = ± tan θ (A.27) (A.28) (A.29) (A.30) sin( π 2 + θ) = cosθ (A.3) ) cos( π + θ) = sinθ 2 (A.32) tan( π 2 + θ) = tan θ a sin A = b sin B = c sin C = 2R a 2 = b 2 + c 2 2bc cos A b 2 = c 2 + a 2 2ca cos B c 2 = a 2 + b 2 2ab cos C (A.33) (A.34) (A.35) (A.36) (A.37) sin(α ± β) = sinα cosβ ± cosα sinβ cos(α ± β) = cosα cosβ sinα sinβ tanα ± tanβ tan(α ± β) = tanα tanβ (A.38) (A.39) (A.40) ) ABC R

G 47 sin2θ = 2sinθ cosθ cos2θ = cos 2 θ sin 2 θ tan2θ = 2tanθ tan 2 θ sin 2 θ 2 = cosθ 2 cos 2 θ 2 = + cosθ 2 tan 2 θ 2 = cosθ + cosθ sinα + sinβ = 2sin α + β cos α β 2 2 sinα sinβ = 2cos α + β 2 cosα + cosβ = 2cos α + β 2 cosα cosβ = 2sin α + β 2 sin α β 2 cos α β 2 sin α β 2 (A.4) (A.42) (A.43) (A.44) (A.45) (A.46) (A.47) (A.48) (A.49) (A.50) y = sin x < > x = sin y (A.5) ( x, π 2 y π 2 ) y = cos x < > x = cos y (A.52) ( x, 0 y π) y = tan x < > x = tan y (A.53) ( x, π 2 y π 2 )

48 A H y y = e x y = x y = log e x 0 x A.5 e x n = x n (A.54) x m x n = x m+n (A.55) (x m ) n = x m n (A.56) n x = x n (A.57) a 0 = (A.58) a b = c < > b = log a c (A.59) a > 0, a, b > 0, c > 0, d > 0, d

H 49 log a (b c) = log a b + log a c log a b c = c log a b c log a b = log a c log a b log a b = log a b log a b = log d b log d a log a a = log a = 0 (A.60) (A.6) (A.62) (A.63) (A.64) (A.65) (A.66)

50 A I J x lim x 2 3 = 4 y = x = 0 x lim x ±0 x = ± ± x 2 lim x x = lim x = lim x (x + ) = 2 (x + )(x ) x x 2 lim x x = lim ( + h) 2 h 0 ( + h) ( + 2h + h 2 ) = lim h 0 h 2h + h 2 = lim h 0 h 2 + h = lim h 0 = lim 2 + h h 0 = 2 + 0 = 2

K ɛ δ K ɛ δ 5 x a lim f(x) = b x a ɛ δ ɛ > 0, δ > 0s.t. x R, 0 < x a < δ f(x) b < ɛ f or all ɛ > 0, there exist δ > 0 such that for all x R, 0 < x a < δ implies f(x) b < ɛ ɛ δ 0 < x a < δ x f(x) b < ɛ 2) 2) x x( ) x x a b s.t. a = a + b a a = a + b b s.t. such that implies imply implies if then ( ) implies

52 A lim (2x ) = 5 x 3 ɛ δ 0 < x 3 < δ x (2x ) 5 < ɛ ɛ δ = ɛ 2 δ ɛ = 2 δ = 2 2 = :x = 4.0, x 3 = 4 3 = δ =, (2x ) 5 = 8 6 = 2 ɛ = 2 :x = 3.9, x 3 = 3.9 3 = 0.9 < δ =, (2x ) 5 = 7.8 6 =.8 < ɛ = 2 ɛ = 0.2 δ = 0.2 = 0. 2 :x = 3.0, x 3 = 3. 3 = 0.9 δ = 0., (2x ) 5 = 6.2 6 = 0.2 ɛ = 2 :x = 3.09, x 3 = 3.09 3 = 0.09 < δ = 0., (2x ) 5 = 6.8 6 = 0.8 < ɛ = 0.2 ɛ = 0.02 δ = 0.02 = 0.0 2 x = 3.00 x = 3.009 OK :x = 3.00, x 3 = 3.0 3 = 0.0 δ = 0.0, (2x ) 5 = 6.02 6 = 0.02 ɛ = 0.02 :x = 3.009, x 3 = 3.009 3 = 0.009 < δ = 0.0, (2x ) 5 = 6.08 6 = 0.08 < ɛ = 0.02 ɛ δ x

B A y = f(x) f(x) x = p x = p f (x) f(x) f(p) lim x p x p (B.) P P(p, f(p)) y f(x) f(p) P p x x B.

54 B f(x) = x 2 x = f(x) f() x 2 lim = lim x x x x = lim (x )(x + ) x x = lim x (x + ) (B.2) x = f(x) = x x = 0 x 0 lim x ±0 x 0 = ± (B.3) x = 0

B B 55 x f (x) y = f(x) dy dx d dx f(x) {f(x)} f (x) y ẏ D x y Df(x) y = f(x) x x y y {f(x)} = y lim x 0 x = lim x 0 f(x + x) f(x) x x h B.2 y f(x + h) y f(x) x x x + h x B.2 {f(x)} = lim h 0 f(x + h) f(x) h (B.4) f(x)

56 B C C.. (a) = 0 ) 2. (x) = 2) 3. (a x) = a 3) 4) 5) 4. (x 2 ) = 2x 5. ( x) = 2 6) 7) x (x n ) = n x n (B.5) ) 2) 3) (a) = (x) = a a lim = 0 (a : ) h 0 h {(x + h)} x lim = h 0 h (a x) = (a) x + a (x) = a (a : ) 4) (x 2 ) = (x x) = (x) x + x (x) = 2x 5) 6) 7) (x 2 ) (x + h) 2 x 2 x 2 + 2hx + h 2 x 2 = lim = lim h 0 h h 0 h 2hx + h 2 = lim = lim (2x + h) = 2x h 0 h h 0 ( x) = lim h 0 x + h x h = lim = h 0 x + h + x 2 x ( x + h x)( x + h + x) = lim h 0 h ( x + h + x) ( x) = (x 2 ) = 2 x 2 = 2 x

C 57 C.2 {f(x) ± g(x)} = f (x) ± g (x) (B.6) 8) {f(x) g(x)} = f (x) g(x) + f(x) g (x) (B.7) 9) 8) {f(x) ± g(x)} {f(x + h) ± g(x + h)} {f(x) ± g(x)} = lim h 0 h {f(x + h) f(x)} ± {g(x + h) g(x)} = lim h 0 h = lim h 0 {f(x + h) f(x)} = f (x) ± g (x) h {g(x + h) g(x)} ± lim h 0 h 9) {f(x) g(x)} {f(x + h) g(x + h)} {f(x) g(x)} = lim h 0 h {f(x + h) g(x + h) f(x + h)g(x)} + {f(x + h)g(x) f(x)g(x)} = lim h 0 h = lim h 0 {f(x + h) f(x)} = f (x) g(x) + f(x) g (x) h {g(x + h) g(x)} g(x) + lim f(x + h) h 0 h

58 B j ff f(x) = f (x) g(x) f(x) g (x) g(x) g(x) 2 (B.8) j ff = g (x) g(x) g(x) 2 (B.9) 0) ) {f(g(x))} = f (u) g (x) u = g(x) (B.0) 0) f(x) g(x) «= lim h 0 f(x + h) g(x + h) f(x) «g(x) f(x + h) g(x) f(x) g(x + h) = lim h h 0 g(x + h) g(x) h f(x + h) g(x) f(x)g(x) {f(x) g(x + h) f(x)g(x)} = lim h 0 g(x + h) g(x) h = lim h 0 f(x + h) f(x) {g(x + h) g(x)} h = f (x) g(x) f(x) g (x) g(x) 2 g(x) f(x) lim h 0 g(x + h) g(x) {g(x + h) g(x)} h ) {f(g(x))} f(g(x + h)) f(g(x)) = lim h 0 h f(g(x + h)) f(g(x)) g(x + h) g(x) = lim h 0 g(x + h) g(x) h f(g(x + h)) f(g(x)) g(x + h) g(x) = lim lim h 0 g(x + h) g(x) h 0 h d = dg(x) f(g(x)) d dy g(x) = dx du du dx = f (u) g (x)

C 59 C.3 sin x lim h 0 x = (B.) (sin x) = cos x (B.2) 2) (cos x) = sin x (B.3) 3) (tan x) = cos 2 x (B.4) 2) (sin x) sin(x + h) sin x = lim h 0 h (x + h) + x 2 cos sin = lim 2 h 0 h 2 cos 2x + h sin h = lim 2 2 h 0 h h = lim h 0 cos(x + h sin 2 ) 2 h 2 = cos x (x + h) x 2 3) (cos x) = (sin( π 2 x)) = cos( π 2 x) ( π 2 x) = cos( π 2 x) ( ) = sin x u = π 2 x

60 B C.4 f(x) = g(y) (B.5) 4) 4) y = x ( x) = x 2 = 2 x 2 = 2 x y = x x = y 2 ( x) = (y 2 ) = 2 y = 2 x

C 6 sin x + cos x = π 2 (B.6) (sin x) = x 2 ( < x < ) (B.7) 5) (cos x) = x 2 ( < x < ) (B.8) (tan x) = + x 2 (B.9) 5) y = sin x x = sin y (sin x) = (sin y) = cos y = q sin 2 y = p x 2 x = sin y d dx x = d dx sin y = d dy sin y dy dx = cos y dy dx dy dx = cos y

62 B C.5 e y = e x y = 2 x x = 0 y = 3 x x = 0 y = e x x = 0 e 2.782882 y y = e x y = x + 0 x B.3 e e e = lim x + x «x (B.20) e = lim ( + x) x x 0 (B.2) e x lim x 0 x = (B.22) (e x ) = e x (B.23) 6) 6) d dx ex e x+h e x = lim = e x e h lim = e x h 0 h h 0 h

C 63 C.6 lim x 0 ( + x) x = e (log e x ) = x (B.24) (B.25) 7) 7) x > 0 (log e x) log = lim e (x + h) log e x h 0 h «x + h log e x = lim h 0 h = lim h 0 h log e + h «x = lim h 0 log e + h «h x 8 >< = lim h 0 log e + h «x 9 >= x h >: x >; = lim h 0 x log e + h «h x x = lim h 0 x log e e = h 0 lim x = x x < 0 u = x (u > 0) y = log e x = log e u = log e u y = (log e u) ( x) = u ( ) = x ( ) = x

64 B C.7 8) 9) (x a ) = ax a (a ) (B.26) (a x ) = a x log e a (a > 0) (B.27) 8) y = x a log e y = log e x a = a log e x 9) x d dx (log e y) = d dx (a log e x) d dy (log dy e y) dx = a d dx (log e x) dy y dx = a x dy dx = a y x = a xa x = axa y = a x log e y = log e a x = x log e a x d dx (log e y) = d dx (x log e a) d dy (log dy e y) dx = d dx x(log e a) dy dx = log e a y dy dx = y log e a = ax log e a

C 65 C.8 f(x) [a, b] (a, b) f(a) = f(b) f (c) = 0 c (a, b) (B.28) f(x) [a, b] (a, b) f(b) f(a) b a = f (c) c (a, b) (B.29) f(x), g(x) [a, b] (a, b) g f(b) f(a) (x) 0 g(b) g(a) = f (c) g (c) c (a, b) (B.30) C.9 f(x), g(x) lim f(x) = lim x a x a g(x) = 0( = ) f (x) lim x a g (x) = c lim f(x) x a g(x) = c (B.3)

66 B f(x) n c f(b) = f(a) + f (a)(b a) + f (a) (b a) 2 + 2! + f (n ) (a) (n )! (b a)n + R n R n = f (n) (c) (b a) (a < c < b) n! (B.32) f(x), f (x), f (x),, f (n ) (x), f (n) (x) x = a f(x) = f(a) + f (a)(x a) + f (a) (x a) 2 + 2! + f (n ) (a) (n )! (x a)n + n, R n 0 (B.33) f(x), f (x), f (x),, f (n ) (x), f (n) (x) x = 0 f(x) = f(0) + f (0)x + f (0) x 2 + + f (n) (0) x n + 2! (n)! n, R n 0 (B.34)

C A f(x) {F (x)} = f(x) (C.) F (x) f(x) f(x) F (x) F (x) + C f(x) F (x) + C = f(x)dx (C ) (C.2) b f(x)dx = [F (x)] b a a = F (b) F (a) (C.3)

68 C B y y = x 0 x 0 x x 2 x 3 x n x n x C. y = x x C. x 0 b y = x lim n X n i=0 x i b n = lim n b n n X i=0 x i = lim n b n 2 (x 0 + x n ) n b = lim n n 2 x n n b = lim n n 2 (x n b n ) n b = lim n n 2 (x n n b) b = lim n n (b n b) 2 =! 2 lim b 2 b2 = n n 2 b2 y = x F (x) = j ff 2 x2 + C = x F (x) = 2 x2 + C = xdx b 0» b xdx = 2 x2 = 2 b2 2 02 = 2 b2 0

C 69 C C. kf(x)dx = k f(x)dx (C.4) f(x) ± g(x)dx = f(x)dx ± g(x)dx (C.5)

70 C x n dx = n + xn+ + C (n ) (C.6) C dx = x (C.7) x dx = log e x (C.8) 3 xdx 2 = 3 x 2 = 2 3 x x (C.9) sin xdx = cos x (C.0) cos xdx = sin x (C.) tan xdx = log e cos x cos 2 dx = tan x x (C.3) x 2 dx = sin x x 2 dx = cos x (C.2) (C.4) (C.5) + x 2 dx = tan x (C.6) e x dx = e x (C.7) a x dx = ax log a (a > 0, a ) (C.8)

C 7 C.2 u = ϕ(x) f(ϕ(x)) ϕ (x)dx = f(u)dx (C.9) f(x)dx = F (x) f(ax + b)dx = F (ax + b)dx (a 0) (C.20) a f(x) a f (x)dx = a + f(x)a+ (a ) (C.2) f (x) f(x) dx = log e f(x) (C.22) ( a 0) (ax + b) a dx = a a + (ax + b)a+ (a ) (C.23) (ax + b) dx = a log e ax + b (C.24) sin axdx = cos ax a (C.25) cos axdx = sin ax a (C.26) e ax dx = a eax (C.27) C.3 f (x) g(x)dx = f(x) g(x) f(x) g (x)dx (C.28)

72 C C.4 x a dx = log e x a (C.29) x 2 + dx = tan x (C.30) 2x x 2 + a dx = log e x2 + a (C.3) x 2 + a 2 dx = x a tan a (a 0) (C.32) tan x 2 = t sin x = 2t + t 2 (C.33) cos x = t2 + t 2 (C.34) dx = 2 dt (C.35) + t2

C 73 C.5 b a f(x)dx = a a b a f(x)dx = 0 f(x)dx (C.36) (C.37) b a f(x) + g(x)dx = b a b f(x)dx + g(x)dx a (C.38) b a b kf(x)dx = k f(x)dx a (k ) (C.39) b a c c f(x)dx + f(x)dx = f(x)dx b a (C.40) [a, b] f(x) g(x) b a b a f(x)dx f(x)dx b a b a g(x)dx f(x)dx u = ϕ(x) [α, β] a = ϕ(α), b = ϕ(β) b a β α f(ϕ(x)) ϕ(x)dx = b f (x) g(x)dx = f(x) g(x) a f(u)du b a f(x) g (x)dx (C.4) (C.42) (C.43) (C.44)

D A α alpha B β beta Γ γ gamma δ delta E ɛ epsilon ζ zeta H η eta Θ θ theta I ι iota K κ kappa Λ λ lambda M µ mu N ν nu Ξ ξ xi O o omicron Π π pi P ρ rho Σ σ sigma T τ tau Υ υ upsilon Φ φ, ϕ phi X χ chi Ψ ψ psi Ω ω omega ℵ

E e 2.78288284 π 3.45926535 2.44235623 3.7320508075 5 2.2360679774 log e 2 0.69347805 log e 3.098622886 log e 0 2.3025850929

RC, 24, 2, 6, 6, 3, 4, 7 ɛ δ, 5, 23, 7, 36, 20, 5, 2, 5, 9, 24, 4, 43, 23, 8, 50, 9, 4 gnuplot, 35, 36, 9, 23, 65, 23, 24, 33, 44, 46, 9, 48 Java, 35, 36 Java, 35, 24, 33, 20,, 8, 6, 35 step, 3 step2, 3 step3, 3 step4, 3 step5, 4 step6, 4 step7, 4 step8, 4, 67, 8, 67, 69, 53, 9, 48, 3, 2, 23, 68, 66, 66, 9, 23, 23, 5, 20, 55

80, 5, 3, 4 Torricelli, 2, 42, 53, 53, 55, 56,,, 8, 65, 8, 4, 24, 26, 33, 66, 6, 3, 20, 30, 3, 29, 33, 30, 30, 3, 35, 35, 3 2, 4 3, 7, 65, 65

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