This document contains lecture notes on calculus topics including:
1) Derivatives of functions including the chain rule and partial derivatives
2) Equations relating supply and demand and solving systems of equations using Jacobians
3) The definition of a Jacobian as the determinant of the matrix of all first-order partial derivatives of a vector-valued function.
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. 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. 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