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# 経済数学II 「第７章 微分法とその比較静学への応用」

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### 経済数学II 「第７章 微分法とその比較静学への応用」

1. 1. shito@seinan-gu.ac.jp 2020 5 26 • • • • 1 1 (1) y = f(x) = cxn dy dx = f (x) = cnxn−1 (2) (a) y = x19 w = 3u−1 (b) d dx (−x−4 ) (3) y = f(x) = k dy dx = f (x) = 1
2. 2. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 (1) pp.182–187 (2) 7.1 1–3 2 (1) y = 3x2 + 9x12 ↔ dy dx = d dx [f(x) ± g(x)] = df(x) dx ± dg(x) dx = f (x) ± g (x) (2) y = (2x + 3)(3x2 ) d dx [f(x)g(x)] = f(x) dg(x) dx + df(x) dx g(x) = f(x)g (x) + f (x)g(x) d dx (2x + 3)(3x2 ) = d dx [f(x)g(x)h(x)] = f (x)g(x)h(x) + f(x)g (x)h(x) + f(x)g(x)h (x) pp.192–195 Revenue = R = PQ = AR = R Q = MR = dR dQ II 2
3. 3. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 (3) d f(x) g(x) dx = d dx f(x) g(x) = f (x)g(x) − f(x)g (x) g(x)2 d dx 4x x2 + 1 = (1) pp.187–198 (2) 7.2 1–8 3 (1) The Chain Rule z = f(y) y = g(x) z = f(g(x)) dz dx = dz dy dy dx = f (y)g (x) dx g −→ dy f −→ dz (2) • y = f(x) 1 1 f • y = f(x) : x = f−1 (y) 1/f • 1 1 ⇐⇒ x1 < x2 =⇒ f(x1) < f(x2) x1 < x2 =⇒ f(x1) > f(x2) II 3
4. 4. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 y x x y dy / dx f f −1 1 対 1 1 対 1 でない y x 逆関数 x = f−1 (y) dx dy = 1 dy/dx y = f(x) x = f−1 (y) dy/dx x = f−1 (y) dx dy = 1 dy/dx : y = x1/3 y (1) pp.199–204 (2) 7.3 1–6 II 4
5. 5. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 4 (partial derivative) 1 1 2 (1) y = f(x1, x2, x3, · · · , xn) x1 ∆x1 ∆y ∆x1 = f(x1 + ∆x1, x2, x3, · · · , xn) − f(x1, x2, · · · , xn) ∆x1 lim ∆x1→0 ∆y ∆x1 ≡ (2) f(x,y) = 0.5*log(x)+0.5*log(y) 0 2 4 6 8 100 2 4 6 8 10 0 0.5 1 1.5 2 2.5 0 2 4 6 8 100 2 4 6 8 10 0 0.5 1 1.5 2 2.5 II 5
6. 6. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 0 2 4 6 8 100 2 4 6 8 10 0 0.5 1 1.5 2 2.5 (3) y = f(x1, x2) = 3x2 1 + x1x2 + 4x2 2 ∂y ∂x1 ≡ f1 = ∂y ∂x2 ≡ f2 = (1) pp.204–209 (2) 7.4 1–4 u 5 (1) Qd = a − bP (a, b > 0) Qs = −c + dP (c, d > 0) Qd = Qs II 6
7. 7. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 ¯Q = Q(a, b, c, d) = ¯P = P(a, b, c, d) = ∂ ¯P/∂a ∂ ¯Q/∂b ∂ ¯P ∂a = ∂ ¯Q ∂b = (2) 45 Y = C + I0 + G0 C = α + β(Y − T) T = γ + δY 0 < β, δ < 1 ⇓ ¯Y = = ¯C = = ¯T = = • II 7
8. 8. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 • • (1) pp.209–214 (2) 7.5 1–2 (3) 45 6 (Jacobian) :::::::: ⇓ |A| 1 Y1 = 2x1 + 3x2 2 Y2 = 4x2 1 + 12x1x2 + 9x2 2 2 1 2 2 |A| = 0 |A| = 0 1 1 2 Jacobian |J| ≡ ∂(y1, y2) ∂(x1, x2) ≡ ∂y1 ∂x1 ∂y1 ∂x2 ∂y2 ∂x1 ∂y2 ∂x2 = |J| = 0 II 8
9. 9. www.seinan-gu.ac.jp/˜shito 2020 5 26 10:26 y1 = f1 (x1, x2, · · · , xn) fi y2 = f2 (x1, x2, · · · , xn) fi i . . . . . . . . . . . . . . . . . . . . . i yn = fn (x1, x2, · · · , xn) |J| ≡ ∂(y1, y2, · · · , yn) ∂(x1, x2, · · · , xn) ≡ ∂y1 ∂x1 ∂y1 ∂x2 . . . ∂y1 ∂xn . . . . . . . . . . . . . . . . ∂yn ∂x1 ∂yn ∂x2 . . . ∂yn ∂xn = 0 4 a11x1 + a12x2 + · · · + a1nxn = d1 a21x1 + a22x2 + · · · + a2nxn = d2 ... ... ... ... am1x1 + am2x2 + · · · + amnxn = dm Jacobian |A| |A| = 0 Jacobian (1) pp.216–219 (2) 7.6 1 II 9
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May. 30, 2020

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