1. 2/ 121
a x = a t 3/ 121
a x = a t 4/ 121
a > 0 t a t = a t t {}}{ a a a t 5/ 121
a t+s = = t+s {}}{ a a a t s {}}{{}}{ a a a a = a t a s (a t ) s = s {}}{ a t a t = a ts 6/ 121
a > 0 t a 0 t t = 0 + t a t = a 0+t = a 0 a t a t 0 a 0 = 1 7/ 121
t 0 = t + ( t) 1 = a 0 = a t+( t) = a t a t a t 0 a t = 1 a t 8/ 121
a = 3 3 4 = 3 3 3 3 = 81 3 0 = 1 3 4 = 1 3 4 = 1 81 9/ 121
t = m n m n m = m n n ( ) a m = a m n n = a m n n n a m n = n a m 10/ 121
a = 3 3 2 1 2 3 3 1 2 = 3 3 3 2 = 3 3 = 3 3 3 8 5 = 3 1+ 3 5 = 3 5 3 3 = 3 5 27 11/ 121
t a = 3 t (1) (2) (3) 12/ 121
x = 3 t x 3 1 = 3 3 2 = 9 3 3 = 27 20 10 3 2 1 O 1 2 3 t 13/ 121
x = 3 t x 3 1 = 3 3 2 = 9 3 3 = 27 20 10 3 2 1 O 1 2 3 t 14/ 121
x = 3 t 3 0 = 1 x 3 1 = 1 3 20 3 2 = 1 9 10 3 3 = 1 27 3 2 1 O 1 2 3 t 15/ 121
x = 3 t 3 0 = 1 x 3 1 = 1 3 20 3 2 = 1 9 10 3 3 = 1 27 3 2 1 O 1 2 3 t 16/ 121
x = 3 t 3 1 2 = 3 x 1.73 3 1 4 = 4 3 1.32 20 10 1 2 3 2 1 O 3 t 17/ 121
x = 3 t 3 1 2 = 3 x 1.73 3 1 4 = 4 3 1.32 20 10 1 2 3 2 1 O 3 t 18/ 121
x = 3 t 20 x t 10 3 2 1 O 1 2 3 t 19/ 121
x x = a t x = a t a t+s = a t a s 1 (a t ) s = a ts O t 20/ 121
2. 21/ 121
t x x t 22/ 121
x x = f(t) t = f(t 2) f(t 1 ) t 2 t 1 t 1 = t t 2 = t + h x t = f(t + h) f(t) (t + h) t = f(t + h) f(t) h 23/ 121
x = f(t) = a t x t = f(t + h) f(t) h = at+h a t = at a h a t h h = at (a h 1) = a t ah 1 h h 24/ 121
a t ah 1 h ah 1 h h 0 x = a t t = 0 a h x = a h 1 1 x O x = a t t = h 25/ 121 h t
a t ah 1 h ah 1 h h 0 x = a t t = 0 x x = a t a h 1 x = a h 1 t = h O h 26/ 121 t
a t ah 1 h ah 1 h h 0 x = a t t = 0 1 x Oh x = a t 27/ 121 t
x x = a t a h 1 h 1 O a h 1 t = 0 h t 28/ 121
x x = a t a h 1 h 1 O a h 1 t = 0 h t 29/ 121
x x = a t a h 1 h 1 O a h 1 t = 0 h t 30/ 121
x x = a t 1 O a h 1 t = 0 h t = 0 t 31/ 121
x x = a t 1 O x = a t a h 1 t = 0 h t = 0 1 a t 32/ 121
x x = a t 1 O a h 1 t = 0 h a h 1 t = 0 1 h t a e 33/ 121
x x = e t 1 O e h 1 t = 0 = 1 h x = e t t = 0 1 e t 34/ 121
x = e t x = e t x t = et+h e t h = et (e h 1) h = e t eh 1 h e t 1 = e t = x 35/ 121
x = e t x = e t x t = et+h e t h = et (e h 1) h = e t eh 1 h e t 1 = e t = x 36/ 121
x = e t x = e t dx dt = et = x 37/ 121
John Napier 1550 1617 4 e = 2.718281828 e (Euler) e 38/ 121
John Napier 1550 1617 4 e = 2.718281828 e (Euler) e 39/ 121
3. 40/ 121
x = e t x = e t +C 41/ 121
x = e t e t e t e t + C e t 42/ 121
x = e t e t e t e t + C e t 43/ 121
x = e t x = e t e t e t e t e t + C 44/ 121
x = e t x = e 2t x x = e 2t x = e 2t t = 0 1 1 O x = e t 1 t 45/ 121
x = e t x x = e 2t x = e t t = 0 1 e 1 x = e t 1 O t 46/ 121
t = 0 x x = e 2t x = e 2t 1 x = e t 1 O t 47/ 121
t = 0 x x = e 2t x = e 2t 1 x = e t 1 O t 48/ 121
x = e kt ke kt 49/ 121
x = e 2t k = 2 ke kt 2e 2t x = e 2t t = 0 2e 2t t = 0 2 50/ 121
x = e kt e kt ke kt e kt 1 k ekt + C 51/ 121
4. 52/ 121
(1) 3 5 3 3 = 3 5+( 3) = 3 2 = 9 a t a s = a t+s (2) ( 2 3) 2 = 2 3 ( 2) = 2 6 = 64 (a t ) s = a t s 53/ 121
(1) 3 5 3 3 = 3 5+( 3) = 3 2 = 9 a t a s = a t+s (2) ( 2 3) 2 = 2 3 ( 2) = 2 6 = 64 (a t ) s = a t s 54/ 121
(1) 3 5 3 3 = 3 5+( 3) = 3 2 = 9 a t a s = a t+s (2) ( 2 3) 2 = 2 3 ( 2) = 2 6 = 64 (a t ) s = a t s 55/ 121
(1) 3 5 3 3 = 3 5+( 3) = 3 2 = 9 a t a s = a t+s (2) ( 2 3) 2 = 2 3 ( 2) = 2 6 = 64 (a t ) s = a t s 56/ 121
(1) 3 5 3 3 = 3 5+( 3) = 3 2 = 9 a t a s = a t+s (2) ( 2 3) 2 = 2 3 ( 2) = 2 6 = 64 (a t ) s = a t s 57/ 121
(1) e 4t 4e 4t e kt ke kt (2) e kt + at 2 + bt + c ke kt + 2at + b (3) e 8t 1 8 e8t + C e kt 1 k ekt + C 58/ 121
(1) e 4t 4e 4t e kt ke kt (2) e kt + at 2 + bt + c ke kt + 2at + b (3) e 8t 1 8 e8t + C e kt 1 k ekt + C 59/ 121
(1) e 4t 4e 4t e kt ke kt (2) e kt + at 2 + bt + c ke kt + 2at + b (3) e 8t 1 8 e8t + C e kt 1 k ekt + C 60/ 121
(1) e 4t 4e 4t e kt ke kt (2) e kt + at 2 + bt + c ke kt + 2at + b (3) e 8t 1 8 e8t + C e kt 1 k ekt + C 61/ 121
(1) e 4t 4e 4t e kt ke kt (2) e kt + at 2 + bt + c ke kt + 2at + b (3) e 8t 1 8 e8t + C e kt 1 k ekt + C 62/ 121
(1) e 4t 4e 4t e kt ke kt (2) e kt + at 2 + bt + c ke kt + 2at + b (3) e 8t 1 8 e8t + C e kt 1 k ekt + C 63/ 121
a > 0, b > 0 (1) ( a 2 a 3) ( ) 4 = a 20 (5) a 3 2 2 b 0 = a 3 (2) 3 2 4 3 3 3 = 9 (3) 5 1 4 3 5 3 = 5 (4) a 1 2 1 2 a 3 = a 6 (6) 8 1 2 8 1 6 = 2 ( ) (7) a 2 5 6 3 b 6 = a 4 b 5 (8) ( ) 25 2 3 4 3 = 1 5 64/ 121
a > 0, b > 0 (1) ( a 2 a 3) ( ) 4 = a 20 (5) a 3 2 2 b 0 = a 3 (2) 3 2 4 3 3 3 = 9 (3) 5 1 4 3 5 3 = 5 (4) a 1 2 1 2 a 3 = a 6 (6) 8 1 2 8 1 6 = 2 ( ) (7) a 2 5 6 3 b 6 = a 4 b 5 (8) ( ) 25 2 3 4 3 = 1 5 65/ 121
(1) e kt ke kt (2) e 3t + 2t + 4 3e 3t + 2 (3) e 5t 1 5 e5t + C (4) e kt 1 k e kt + C 66/ 121
(1) e kt ke kt (2) e 3t + 2t + 4 3e 3t + 2 (3) e 5t 1 5 e5t + C (4) e kt 1 k e kt + C 67/ 121
5. 68/ 121
x = e t t x x x = e t x t x 1 O t t 69/ 121
x > 0 x t x x = e t x x > 0 x t 1 O t t 70/ 121
x > 0 x x = e t t x t x x = e t x 1 O t t x t = log x 71/ 121
t = log x x = e t t = log x = log e t x = e t = e log x 72/ 121
t = log x = log e t t = 1 e 1 = e log e = 1 t = 0 e 0 = 1 log 1 = 0 73/ 121
74/ 121
t = log x s = log y x = e t y = e s u = log xy xy = e u e u = xy = e t e s = e s+t u = s + t log xy = log x + log y 75/ 121
t = log x x = e t s = log x m x m = e s e s = x m = (e t ) m = e m t s = m t log x m = m log x 76/ 121
log e = 1 log 1 = 0 log xy = log x + log y log x m = m log x 77/ 121
6. 78/ 121
79/ 121
dt dx = 1 dx dt 80/ 121
x = e t dx dt = et = x t = log x dt dx = 1 dx dt = 1 e t = 1 x 81/ 121
x = e t dx dt = et = x t = log x dt dx = 1 dx dt = 1 e t = 1 x 82/ 121
x = e t dx dt = et = x t = log x dt dx = 1 dx dt = 1 e t = 1 x 83/ 121
log x 1 x log x + C 1 x 84/ 121
log x 1 x log x + C 1 x 85/ 121
1 x log x 1 x 1 x log x + C 86/ 121
7. 87/ 121
log 2 = a, log 3 = b a, b (1) log 24 = log 2 3 3 = log 2 3 +log 3 = 3 log 2+log 3 = 3a+b (2) log 1 4 = log 1 2 2 = log 2 2 = 2 log 2 = 2a (3) log 2e 3 = log 2 + log e 3 = a + 3 88/ 121
log 2 = a, log 3 = b a, b (1) log 24 = log 2 3 3 = log 2 3 +log 3 = 3 log 2+log 3 = 3a+b (2) log 1 4 = log 1 2 2 = log 2 2 = 2 log 2 = 2a (3) log 2e 3 = log 2 + log e 3 = a + 3 89/ 121
log 2 = a, log 3 = b a, b (1) log 24 = log 2 3 3 = log 2 3 +log 3 = 3 log 2+log 3 = 3a+b (2) log 1 4 = log 1 2 2 = log 2 2 = 2 log 2 = 2a (3) log 2e 3 = log 2 + log e 3 = a + 3 90/ 121
log 2 = a, log 3 = b a, b (1) log 24 = log 2 3 3 = log 2 3 +log 3 = 3 log 2+log 3 = 3a+b (2) log 1 4 = log 1 2 2 = log 2 2 = 2 log 2 = 2a (3) log 2e 3 = log 2 + log e 3 = a + 3 91/ 121
(1) log x = 3t x = e 3t 1 (2) log 3x log x x 1 log 3x = log x + log 3 x (3) 3 x 1 x log x + C 3 1 x 3 log x + C = log x3 + C 92/ 121
(1) log x = 3t x = e 3t 1 (2) log 3x log x x 1 log 3x = log x + log 3 x (3) 3 x 1 x log x + C 3 1 x 3 log x + C = log x3 + C 93/ 121
(1) log x = 3t x = e 3t 1 (2) log 3x log x x 1 log 3x = log x + log 3 x (3) 3 x 1 x log x + C 3 1 x 3 log x + C = log x3 + C 94/ 121
(1) log x = 3t x = e 3t 1 (2) log 3x log x x 1 log 3x = log x + log 3 x (3) 3 x 1 x log x + C 3 1 x 3 log x + C = log x3 + C 95/ 121
(1) log x = 3t x = e 3t 1 (2) log 3x log x x 1 log 3x = log x + log 3 x (3) 3 x 1 x log x + C 3 1 x 3 log x + C = log x3 + C 96/ 121
(1) log x = 3t x = e 3t 1 (2) log 3x log x x 1 log 3x = log x + log 3 x (3) 3 x 1 x log x + C 3 1 x 3 log x + C = log x3 + C 97/ 121
log 2 = a, log 3 = b (1) log 144 = 4a + 2b (2) log 1 6 = a b (1) log x 2 = 4t x = e 2t (2) log x 2 2 x (3) k x k log x + C = log xk + C 98/ 121
log 2 = a, log 3 = b (1) log 144 = 4a + 2b (2) log 1 6 = a b (1) log x 2 = 4t x = e 2t (2) log x 2 2 x (3) k x k log x + C = log xk + C 99/ 121
8. 100/ 121
dx dt = kx 101/ 121
dx dt = kx 1 x t dx x = kdt 102/ 121
dx dt = kx 2 1 x dx = ( k) dt 1 dx x = kdt log x = kt + C 0 C 0 103/ 121
dx dt = kx 3 x = x = e kt+c 0 = e C 0 e kt 1 2 dx x = kdt 1 x dx = ( k) dt = Ce kt log x = kt + C 0 e C 0 = C 104/ 121
dx dt = kx x = Ce kt 1 dx x = kdt 2 1 x dx = ( k) dt log x = kt + C 0 C 3 x = e kt+c 0 = e C 0 e kt = Ce kt 105/ 121
dx dt = kx x = Ce kt 1 dx x = kdt 2 1 x dx = ( k) dt log x = kt + C 0 C 3 x = e kt+c 0 = e C 0 e kt = Ce kt 106/ 121
dx dt = kx + g 107/ 121
dx dt = kx + g 1 dx dt = k ( x g k ) 108/ 121
dx dt = kx + g 2 1 ( dx dt = k x g ) k X = x g k g k 0 109/ 121
dx dt = dx dt 1 ( dx dt = k x g ) k 2 dx dt = dx dt = kx X = x g k 110/ 121
dx dt = kx + g 3 dx = kx dt X = Ce kt X = x g k x = Ce kt + g k 1 dx dt = k ( x g ) k 2 X = x g k dx dt = dx dt = kx 111/ 121
dx dt = kx+g x = Ce kt + g k 1 2 3 dx dt = k ( x g ) k X = x g k dx = dx dt dt dx = kx dt X = Ce kt x = Ce kt + g k C 112/ 121
dx dt = kx+g x = Ce kt + g k 1 2 3 dx dt = k ( x g ) k X = x g k dx = dx dt dt dx = kx dt X = Ce kt x = Ce kt + g k C 113/ 121
dv dt = k m v + g t = 0 v = 0 v m g k 114/ 121
dv dt = k ( v mg ) m k V = v mg k dv = dv dt dt 115/ 121
dv dt = k m V 1 V dv = k m dt 116/ 121
1 V dv = ( k ) dt m log V = k m t + C 0 C 0 117/ 121
V = e k m t+c 0 = e C 0 e k m t C = e C 0 V = Ce k m t V = v mg k 118/ 121
dv dt = k m v + g v = Ce k m t + mg k 119/ 121
t = 0 v = 0 v = mg k e 0 = Ce 0 + mg k C = mg k k m t + mg k = mg k (1 e k m t ) 120/ 121
v mg k v = mg k (1 e k m t ) O t t v mg k 121/ 121