8. 2 1 2 1 2 ma,y u(, y) s.t. p + p y y = m u y y p p y y m u(, y) = y p + p y y = m y ( ) 1 y = (m p ) p y = m p y p p y 2 0 m/p U U() = m p y p p y 2 2 du() d = m p y 2p p y 1
0 = m 2p 1 p = 1/2 p y = 1 m = 1 2 d ( + 1)2 d ( + 1) 2 = 2 + 2 + 1 d d ( + 1)2 = 2 + 2 = 2( + 1) 2 g() 2 f() f() = [g()] 2 = g()g() f f () = [g()g()] = g ()g() + g()g () = 2g()g () 3 4 ((f()) n ) = n(f()) n 1 f () 2 (3 2) 2, (1 5) 4,( 3 + 2) 5 2
3 3.1 y = ( + 1) 2 + 1 2 + 1 ( + 1) 2 2 2 z y z g f() = y g(y) f() = 1 g(y) = y 2 f() = y z = g(f()) z f g f : X Y g : Y Z g f : X Z { (fg)() = f()g() (g f)() = g(f()) 3 y = ( + 1) 2 3
4 f() = ( 2 + 1) 2 g(y) = y 3 + 2 h = g f f g h = g f f g z dz d y dy d y z dz dy 2 dz d = dz dy dy d dz dz dy () = (y) d dy d () (y = f()) y f (g f) = g (y) f () (y = f()) (g f) = g (f()) f () (2) dy dz, dz, dy d dy d (1) 4
(g f) () = g (y) f () ( y = f()) (g f) = g f 5 y = ( + 1) 2 6 f() = ( 2 + 1) 2 g(y) = y 3 + 2 h = g f 7 f() = 3 + 1 g(y) = y 2 h = g f 8 f() = 1 ( 2 +1) 3 3.2 y = f() z = f(y) z 0 z = z y y 0 y 0 z lim 0 y = dz dy 5
( dz d = lim z z 0 = lim 0 y y ) = dz dy dy d ( = lim 0 ) ( z lim y 0 ) y 4 4.1 f : X Y X f( + h) f() lim h 0 h f a > 0 f() = a a +h a lim h 0 h = lim h 0 a a h a h = a lim h 0 a h 1 h = a lim h 0 a h a 0 h = 0 f (0) = lim h 0 a h a 0 h (2) f () = a f (0) (3) e e ( ) ( e = lim 1 + 1 ) n n n 6
e h 1 lim h 0 h = 1 e = 0 1 y y = a + b a b 0 1 e (e ) = e e 1 e rt t 1 t rt e rt (e rt ) = (e y ) (rt) = re rt 9 y = e 2+1 7
10 y = e 3 11 y = e 1/ 12 y = e e f() (e f() ) = f ()e f() 4.2 M r 100% n lim (1 M + r ) kn = Me rn k k n f(t) = e rt f(t) t f (t) = re rt dt t 0 f(t) df dt (t 0) f(t 0 ) 8
df (t dt 0) f(t 0 ) = rert0 e t 0 = r e rt r 13 r 100% 1 1 1 + r 1 t 1 e rt 5 5.1 f : X Y f 1 : Y X y = f() = f 1 (y) 2 f() = 2 ( 0) f 1 (y) = y f 1 1 f() = 2 + 1 9
2 y = 1 2 y 1 2 1/2 (f 1 ) (y) = 1 f () ( f () 0 ) 14 f() = 5 8 15 f() = 5.2 2 (L) Q w MC = w MP L 2 C = wl L Q 1 MC = dc dq = w dl dq dl dq = 1 dq dl 1 L = g(q) g Q 10
MP L = dq dl MC = w MP L 16 F (L) = L C(Q) w 17 MC = 2wQ 5.3 f f 1 f 1 f (f 1 f)() = f 1 (f()) = f 1 (y) = (f 1 f)() = f 1 (f()) = = d d f 1 (f()) = (f 1 ) (y) f () = d d = 1 (f 1 ) (y) f () = 1 ( & y = f()) f () 0 f () (f 1 ) (y) = 1 f () 11
(f 1 ) (y) = 1 f () ( f () 0 ) 5.4 1 f = dy d (f 1 ) = d dy d dy = 1 dy d ( dy d 0) d dy (y) = 1 dy () (f() = y) d d d dy = 1 dy d 18 y = 3 6 6.1 e natural logarithm e ln 12
log e = log = ln y = e = log y = e log (f 1 ) (y) = 1 f () ( f () 0 ) log log y = f() = log = e y f () = (log ) = 1 (e y ) = 1 e y = 1 log (log ) = 1 1 0 0 log 1 log 1 2 2 2 1 1 3 f() = log(+1) = 0 3 > 0 f () = (log( + 1)) = 1 + 1 13
y y = log( + 1) 1: f() = log( + 1) = 0 f (0) = 1/(1 + 0) = 1 1 f(0) = log 1 = 0 1 19 y = log(2 + 1) 20 y = log(1 3) 21 y = log 22 y = 2 log 23 (log ) = 1 log f() (log f()) = f () f() f() 14
6.2 f (log f()) (log f()) = f () f() 24 y = e rt 25 y = 1/ 26 f() = (+a)(+b) f 1 () = f()( + 1 ) (+a) (+b) 6.3 E E = D D = = D P P P D D P E = dd dp dd D E = dp P P D = d log D d log P P 15
d log D dd = 1 D = d log D = dd D d log P = 1 = d log P = dp dp P P d log D dd d log P = D dp P = dd dp P D = E 7 y t f y = f(t) dy dt = y = f (t) f(t) f(t) log f() (log f()) = f () f() d(log f(t)) = dt = f (t) f(t) Y K L A Y = AK α L 1 α α 1 Y log Y = log A + α log K + (1 α) log L 16
Y K L A t Y Y = A A + αk + (1 α)l K L = + α + (1 α) 8 f : X Y a X f(a + h) f(a) lim h 0+ h f a f a f +(a) f(a + h) f(a) lim h 0 0 h f a f a f (a) a b (a < b) 17
[a, b] [a, b] f I = [a, b] (a < b) f (a, b) a b (, b] [a, + ) 27 f() = = 0 1 ( ) a f f (a) = α f +(a) = α f (a) = α 9 1 2 ( ) f [a, b] (a < b) (a, b) f(b) f(a) b a = f (c) c (a, b) 18
2 y A C c B 2: A B 2 2 A B C 0 28 f() = 3 3 2 [0, 3] c 10 km/h f t = f(t) f(t 0 ) = 0 km f(t 1 ) = 1 km 3: t 0 t 1 f(t 1 ) f(t 0 ) t 1 t 0 19
f (t 0 ) f(t 1 ) f(t 0 ) lim t 1 t 0 t 1 t 0 velocity v = ( ) v(t) = f (t) = ( ) = v (t) 2 = f (t) = ( ) = ( ) 11 2 2 2 1 2 ( ) d df d d d 2 f d 2 20
df 2 2 d f () D 2 f() 2 f() = 2 y y = 2 0 4: y = 2 f () = 2, f () = 2 1 y f () = 2 2 0 5: f 1 21
2 1 1 2 2 y 2 f () = 2 0 6: f 2 12 1 2 f () > 0 1 f () > 0 f () > 0 7: f () > 0 1 2 f() = 2 2 22
f () < 0 f () > 0 8: f () < 0 y y = 2 0 9: y = 2 f () > 0 f () > 0 2 f () < 0 1 1 2 g() = 2 2 23
f () > 0 f () < 0 10: f () < 0 f () < 0 f () < 0 11: f () > 0 f () < 0 f () < 0 2 f () = 0 29 y = 3 3 24
y y = 2 1 1 12: y = 2 12.1 30 u = u() = 1. 25
31 y MRS y = dy d = y u = u 2 13 1 / 2 4 f() = y = 3 3 2 + 1 4 y = 3 2 6 = 3( 2) = 0, 2 0 < 0 f () 0 < < 2 f () 2 < f () y = 6 = 6( 1) = 1 f () 0 < 1 f () 1 < f () 0 1 2 y + 0 0 + y 0 + + + y 1 1 3 1: y 26
y 1 2 3 3 13: 3 32 f() = y = 3 + 12 33 Q C(Q) C(Q) = Q 3 4Q 2 + 7Q + 64 14 = 0 1 = a f f (a) = 0 a f () > 0 ( < a) f () < 0 ( > a) 27
= a f f (a) = 0 a f () < 0 ( < a) f () > 0 ( > a) 34 y = 3 3 y = 3 + 12 15 2 y y 0 y = 2 0 y = 2 14: y = 2 y = 2 = 0 0 f ( ) = 0 f ( ) < 0 f ( ) > 0 2 f ( ) = 0 1 28
1 f ( ) = 0 1. f ( ) < 0 2. f ( ) > 0 1. f ( ) = 0 2. f ( ) < 0 3. f( ) y = f() = 2 4 + 4 (0 3) f(2) = 0 f () = 2 4, f (2) = 0 f () = 2 > 0 2 = 0 35 2 f() = ( 0) 36 L Y = F (L) = L p w 29
y y = 2 4 + 4 (0 3) 0 15: y = 2 4 + 4 (0 3) 16 e 2 y = e 2 y y = 2e 2 y = 2e 2 2( 2)e 2 = 2(2 2 1)e 2 e 2 0 16: e 2 30