II (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1

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1 II I r /8 = a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2 (1) n n 2 y = f(x) = e x x = 0 1 f (0) = 1 f(0) = 1 y = x + 1 x 0 e x x r/100 e r/100 (1) 2 (e r/100 ) n = e nr/100 nr 100 log 2 n 100 log 2 r

2 II (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1 f(0) = log 1 = 0 y = g(x) = x x 0 log(1 + x) x (2) log(1 + r/100) r/100 n 100 log 2 r n log 2 = log 10 2/ log 10 e 0.69 n 69 r elasticity ε

3 II ε(p) = = q / p q p = q p p q p ε(p) q p ε(p) = lim p 0 p q = f (p) p q p = = 10 p/p = 10/10000 = 1/ p = = 10 p/p = 10/100 = 1/ P = log e p P p (3) (4) (5) (6) P 1 p (7) p P Q = log e q Q Q 1 q (8) q

4 II P p ε(p) = Q lim P 0 P ε(p) = dq dp = d(log e f(p)) d(log e p) (9) (10) (10) d(log e p) log e p P Q f log e f(p) (10) Q P P p P = log e p p = e P Q = log e f(p) p = e P Q = log e f(p) Q = log e f(e P ) (11) P Q 2 ε(p) = dq dp = 1 f(e P ) f (e P )e P = p f(p) f (p) = p q f (p) (12) (6) R ++ = (0, ) q = f(p) = 1 p (13) ε(p) = p 1/p ( 1)p 2 = 1 (14)

5 II f II I 2 2 z = f(x, y) x y z I 1 2 x y z w = f(u, v) z = f(x, y) 2 2 x-y-z 1 x y z x y z z x x y y 1 2 z x y z α x + β y α β x a y b x x x = a + x y b b + y z f(a, b) f(a + x, b + y) z = f(a + x, b + y) f(a, b) 2 z α x + β y

6 II α β α z = f(x, y) x x = a y = b β z = f(x, y) y x = a y = b x y 1 2 x y 2 3 2, z = 3x + 4y 5, z = x 2 + y 2, z = e x+2y2. 1, 2.,,. 2 z = f(x, y). x y, z., 2, z 2 z

7 II z = f(x, y) (x, y) = (a, b) x, y b x a x = a x = a + x x z z = f(a, b) z = f(a + x, b) z = f(a + x, b) f(a, b) (x, y) = (a, b) x z lim x 0 x (15) x 1 z = f(x, b) x = a z x (a, b) f x(a, b) z x (a, b). y. (a, b) 2 (a, b) 2 x y. z x = f x (x, y), z y = f y (x, y) z x, z y., x( y) x( y). 1 x y x z = xy 2 x y z = (y 2 )x y x y 2 x z z x = y 2 z x = y2

8 II y x 3 5 y z 2xy z y = 2xy z y = 2xy (x, y) = (3, 2) z x (3, 2) = 22 = 4, z (3, 2) = = 12 y z = ax + b cy + d z x = a cy+d, z y = c(ax+b) (cy+d) 2 2. z = x 2 xy + y 2 z x = 2x y, z y = x + 2y 3. z = y 1 + x 2 zx = 2xy (1+x 2 ) 2, z y = 1 1+x 2 4. z = xe x+y z x = (1 + x)e x+y, z y = xe x+y 5. z = xe y ye x z x = e y ye x, z y = xe y e x

9 II z = y log(x 2 + y 2 + 1) z x = 2xy x 2 +y 2 +1, z y = log(x 2 + y 2 + 1) + 2y2 x 2 +y z = f(x, y) = mx + ny + l 1 1 y = f(x) = mx + n 2 y = f(x) = mx + n f x (x, y) = m f y (x, y) = n 1 z = f(x, y) = mx + ny + l x m y n 1 x = 3, y = 2 x x, y y., z z = (m(3 + x) + n(2 + y) + l) (3m + 2n + l) (16) = m x + n y (17) z = (m, n) ( x, y) (18) z = f(x, y) (a, b) (f x (a, b), f y (a, b)) z = f(x, y) (a, b) f (a, b), z (a, b), f(a, b) z(a, b)

10 II ( x, y) X (a, b) A (18) z = f (A) X (19) 1 1 (17) (18) (19) 1 z = f(x, y) = mx + ny + l (x, y) = (a, b) X = A 5 2 z = xy 2, x = 3, y = x x, y y., z z = (3 + x)(2 + y) (20) = 4 x + 12 y + { 3( y) x y + x( y) 2} (21). 3,4,5 x y 2 3 1,2. 1, 1,2., 3. 2 z = xy 2 z 4 x + 12 y (22). z z 4 x + 12 y 2. z = f(x, y) x = a, y = b (, (x, y) = (a, b) ), z α x + β y (23)

11 II z x α, y β. α β 1, 2 2,., (α, β) z = f(x, y) (x, y) = (a, b) f (a, b). 2,. (α, β) (a, b) z = f(x, y), f(a, b). (x, y) X, x y (a, b) A, ( x, y) X, (23) z f (A) X (24), 1.. z = xy 2, z,,. 2 z = f(x, y) (x, y) = (a, b) x, f(a + x, b) f(a, b) lim x 0 x (25). z/ x(a, b) f x (a, b)., x 1 f(x, b) x = a. y. (a, b) 2 (a, b), 1 x, y

12 II z x = f x (x, y), z y = f y (x, y) z/ x, z/ y., x( y) x( y)., z = f(x, y) (x, y) = (a, b). z α x + β y. (26) x,y a,b, y b, x a a + x., y = 0., z α x (27), y b x 1 f(x, b) x = a α., α z = f(x, y) (x, y) = (a, b) x., β z = f(x, y) (x, y) = (a, b) y., z = xy 2 x z/ x = y 2, y z/ y = 2xy., (x, y) = (3, 2) 2, 4 12, (22) (4, 12). 2, x y., (x, y) = (a, b), f (a, b) = f(a, b) = (f x (a, b), f y (a, b)). 2 z = f(x, y) (x, y) =

13 II (a, b) z f (a, b) X (28) = (f x (a, b), f y (a, b)) ( x, y) (29).,, 1. dz = f (a, b) dx (30) = (f x (a, b), f y (a, b)) (dx, dy) (31) = f x (a, b)dx + f y (a, b)dy (32)

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

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