Formulas
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Second derivative test Let r > 0 and assume that all second order derivatives of
the function f (x, y) are continuous at all points (x, y) that are within a distance
r of (a, b). Assume that fx (a, b) = fy (a, b) = 0. Define
D(x, y) = fxx (x, y)fyy (x, y) − [fxy (x, y)]2
ˆ If D(a, b) > 0 and fxx (a, b) > 0, then f (x, y) has a local minimum at (a, b).
ˆ If D(a, b) > 0 and fxx (a, b) < 0, then f (x, y) has a local maximum at (a, b).
ˆ If D(a, b) < 0, then f (x, y) has a saddle point at (a, b).
ˆ If D(a, b) = 0, then we need more information to classify the critical point.
The Simpson’s rule approximation is
Z b
∆x
f (x) dx ≈ [f (x0 ) + 4f (x1 ) + 2f (x2 ) + · · · + 2f (xn−2 ) + 4f (xn−1 ) + f (xn )]
3
a
where n is even, ∆x =
b−a
n ,
and xi = a + i∆x.
Assume that |f (4) (x)|≤ L for all a ≤ x ≤ b. Then the total error introduced by
Z b
L (b − a)5
f (x) dx is not greater than
Simpson’s rule when approximating
.
180
n4
a
First-order linear differential equation Let a and b be constants. The differentiable
dy
function y(x) obeys the differential equation
= a(y − b) if and only if
dx
ax
y(x) = (y(0) − b) e + b.
Selected Taylor series
ex =
sin(x) =
cos(x) =
1
=
1−x
ln(1 + x) =
arctan x =
∞
X
xn
n=0
∞
X
n!
(−1)n
n=0
∞
X
(−1)n
n=0
∞
X
(−1)n
(−1)n
n=0
1 2
1
x + x3 + · · ·
2!
3!
for all −∞ < x < ∞
1
1
1
x2n+1 = x − x3 + x5 − · · ·
(2n + 1)!
3!
5!
for all −∞ < x < ∞
1
x2n
(2n)!
for all −∞ < x < ∞
xn
n=0
∞
X
n=0
∞
X
=1+x+
=1−
1 2
1
x + x4 − · · ·
2!
4!
= 1 + x + x2 + x3 + · · ·
for all −1 < x < 1
xn+1
n+1
=x−
x2 x3 x4
+
−
+ ···
2
3
4
for all −1 < x ≤ 1
x2n+1
2n + 1
=x−
x3 x5
+
− ···
3
5
for all −1 ≤ x ≤ 1
Definitions
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Marshallian and Hicksian Demand Let x be a quantity of good X, let y be
a quantity of good Y , and let u(x, y) be the utility function of these two
goods. Let px resp. py be the unit prices for good X resp. Y .
The function xm (px , py , I) giving the optimal consumption of X to maximize
u(x, y) subject to the budget constraint px x+py y = I is called the Marshallian
demand function.
Let U be the minimum level of utility required by the consumer. The
Hicksian demand function xh (px , py , U ) gives the value of x that minimizes
the cost function f (x, y) = px x + py y subject to the constraint u(x, y) ≥ U .
Surplus Consider a supply curve S(q) and a demand curve D(q) with intersection
point (qe , pe ). The consumer surplus is the area from q = 0 to q = qe under
D(q) and above the line p = pe . The producer surplus is the area from
q = 0 to qe over S(q) and under the line p = pe . The total surplus is the
sum of consumer and producer surplus.
Marginal and Total Cost and Revenue Let TC(q) be the total cost of producing
q units of a particular good. TC(q) is the sum of the fixed cost, TC(0),
and variable costs, TC(q)−TC(0). The marginal cost of production is
d
[TC(q)].
MC(q) =
dq
Let TR(q) be the total revenue collected from q units of output, with
d
TR(0) = 0. The marginal revenue is MR(q) =
[TR(q)] and the unit
dq
TR(q)
price is P(q) =
q
Expected Value, Variance, and Standard Deviation The expected
value of
P
a discrete random variable X with sample space S is E(X) = x · P r(X = x).
S
P
Its variance is V ar(X) = (x − E(X))2 · P r(X = x) = E(X 2 ) − [E(X)]2 .
S
The expected value of a continuous random variable X with PDF f (x) is
R∞
R∞
E(X) =
x · f (x) dx. Its variance is V ar(X) =
(x − E(X))2 · f (x) dx =
−∞
−∞
E(X 2 ) − [E(X)]2 .
The standard deviation of a random variable X is σ(X) =
p
V ar(X).
The Taylor Series for the function f (x) around the centre a is the power series
∞ (n)
P
f (a)
(x − a)n .
n!
n=0