Distance in R3
Acceleration to Velocity if a(t) is acceleration at time t
The distance between the points P1 = (x1 , y1 , z1 ) and P2 =
Z b
(x2 , y2 , z2 ) is
Change in velocity between times a and b =
a(t)dt
p
a
2
2
2
d(P1 , P2 ) = (x2 − x1 ) + (y2 − y1 ) + (z2 − z1 )
Rate of Growth to Total Growth if r(t) is rate of growth
at time t
Equation of a Tangent Plane
Z b
The equation of the plane through a point (x1 , y1 , z1 ) have
slope a in the x-direction and slope b in the y-direction is
Total Growth between times a and b =
r(t)dt
a
z − z1 = a(x − x1 ) + b(y − y1 )
Properties of the Definite Integral
Rb
Rb
Rb
Linear Approximation for (x, y) near (a, b)
1. a [f (x) ± g(x)]dx = a f (x)dx ± a g(x)dx
Rb
Rb
f (x, y) ≈ f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)
2. any constant k, a kf (x)dx = k a f (x)dx
Rb
Rc
Rb
this can be written as ∆f ≈ fx (a, b)∆x + fy (a, b)∆y.
3. if a ≤ c ≤ b, a f (x)dx = a f (x)dx + c f (x)dx
2nd Derivative Test Suppose (a, b) satisfies fx (a, b) = 0
Ra
Rb
and fy (a, b) = 0.
4. b f (x)dx = − a f (x)dx
2
Let D = fxx (a, b)fyy (a, b) − fxy (a, b) . Then,
Ra
5.
f (x)dx = 0
a
• if D > 0 and fxx (a, b) > 0 then f has a local min. at
(a, b)
Fundamental Theorem of Calculus Suppose f (x) is a continuous function on the interval [a, b] and F (x) is an antiderivative of f (x), that is F 0 (x) = f (x) for all x in [a, b].
• if D > 0 and fxx (a, b) < 0 then f has a local max. at
Then,
Z b
(a, b)
f (x)dx = F (b) − F (a)
a
• if D < 0 then f has neither a local min nor local max.
at (a, b)
The Area Function If f (x) is continuous on [a, b], then the
Rt
area function A(t) = a f (x)dx with a ≤ t ≤ b is an antiderivative of f , that is A0 (t) = f (t).
Total Change in Cost The change in cost from changing
• if D = 0 the test is inconclusive.
production from a units to b units is
Method of Lagrange Multipliers To optimize f (x, y) subZ b
Z b
ject to the constraint g(x, y) = c, solve the system
C(b) − C(a) =
C 0 (x)dx =
M C(x)dx
a
fx = λgx
fy = λgy
g(x, y) = c
a
Also, C(x) = C 0 (x)+ constant to be determined.
Total Change in Revenue The change in revenue from changing production from a units to b units is
Z b
Z b
R(b) − R(a) =
R0 (x)dx =
M R(x)dx
R
for x, y, and λ.
Rules for (Basic) Indefinite Integrals
R k
1
1. k 6= 1,
x dx = k+1
xk+1 + c
R 1
2. k = 1,
x dx = ln|x| + c
R kx
3. k 6= 0,
e dx = k1 ekx + c
R
R
4. any k,
kf (x)dx = k f (x)dx
R
R
R
5. [f (x) ± g(x)]dx = f (x)dx ± g(x)dx
a
a
Also, R(x) = R0 (x)+ constant to be determined.
Total Change in Profit The change in profit from changing
production from a units to b units is
Z b
Z b
π(b) − π(a) =
π 0 (x)dx =
[R0 (x) − C 0 (x)]dx
R
a
R
a
0
Also, π(x) = π (x)+ constant to be determined.
Area Between 2 Curves if f (x) ≥ g(x), the the area between f (x) and g(x) over [a, b] is
Z b
Area =
[f (x) − g(x)]dx
Integration by Parts
Z
Z
u · dv = u · v − v · du
a
Definition of the Definite Integral
Z
Average Value of a Function
b
f (x)dx = lim [f (x1 )∆x + f (x2 )∆x + ... + f (xn )∆x]
a
Average value of f (x) on [a, b] =
n→∞
provided the limit on the right exists.
Velocity to Displacement if v(t) is velocity at time t
Z
Displacement between times a and b =
1
b−a
Z
b
f (x)dx
a
Midpoint Rule and Error Bound if n subintervals are used,
and xi is the midpoint of the ith subinterval, then
Z b
f (x)dx ≈ [f (x1 ) + f (x2 ) + ... + f (xn )]∆x
b
v(t)dt
a
a
1
where ∆x =
EM ≤
b−a
n .
The error is bounded by
K(b − a)3
,
24n2
Median of a Continuous Random Variable The median
of a random variable X that has probability density function
f (x) is a number m such that
Z m
1
f (x)dx =
2
−∞
K = max of f 00 (x) on [a, b]
Trapezoidal Rule and Error Bound if n subintervals are
used, and xi is the left end of the ith subinterval, then
Z b
1
1
f (x)dx ≈
f (x1 ) + f (x2 ) + ... + f (xn−1 ) + f (xn ) ∆x
2
2
a
where ∆x =
b−a
n .
Mean of a Continuous Random Variable For a random
variable X with density function f (x), a ≤ x ≤ b, the mean
µ, or expected value E(X), is given by
The error is bounded by
Z
µ = E(x) =
K(b − a)3
ET ≤
,
12n2
xf (x)dx
a
00
K = max of f (x) on [a, b]
Variance of a Continuous Random Variable The variance , V ar(X) or σ 2 , of a random variable X with probability density function f (x), a ≤ x ≤ b and mean µ is
Consumer Surplus and Producer Surplus If qe is the
equilibrium quantity and pe is the equilibrium price, then
Z qe
Z qe
CS =
D(q)dq − pe qe , and P S = pe qe −
S(q)dq
0
b
2
Z
V ar(X) = σ =
b
x2 f (x)dx − µ2
a
0
Standard Deviation of a Continous Random Variable
The
standard deviation, σ, of the random variable X is
nt
• compounded n times per year: F V = P V 1 + nr
, given by
√
p
−nt
σ = σ 2 = V ar(X)
and P V = F V 1 + nr
Integrals of Trigonometric Functions
• continuously compounded: F V = P V ert , and P V =
R
−rt
FV e .
• sin(x)dx = −cos(x) + c
R
PV and FV of a Continuous Income Stream
• cos(x)dx = sin(x) + c
R
Z T
Z T
• tan(x)dx = ln|sec(x)| + c
FV =
S(t)er(T −t) dt,
PV =
S(t)e−rt dt
R
0
0
• cot(x)dx = −ln|csc(x)| + c
R
if S(t) = S is a constant
• sec(x)dx = ln|tan(x) + sec(x)| + c
R
S
S
P V = (1 − e−rT )
F V = (erT − 1),
• csc(x)dx = −ln|cot(x) + csc(x)| + c
r
r
Integrals Involving Inverse Trig. Functions
Useful Limits (for Improper Integrals)
R
√ 1
•
dx = arcsin(x) + c
1−x2
• limb→∞ ebb = 0
R
1
m
dx = arccos(x) + c
• − √1−x
2
• limb→∞ beb = 0 for any m > 0
R 1
•
1+x2 dx = arctan(x) + c
• limb→∞ ln(b)
bm = 0 for any m > 0
Present and Future Value of a Single Investment
• limb→∞
1
xm
• limb→∞
xm
= ∞
= 0 for any m > 0
for any m > 0
Useful Improper Integral
Z
1
∞
1
dx :
xp
(
diverges if p ≤ 1
converges if p > 1
Present Value of a Perpetual Income Stream
Z
PV =
∞
S(t)e−rt dt
0
Properties of Probability Density Functions
1. f (x) ≥ 0 for all x in (a, b) ((a, b) is the domain)
Rb
2. a f (x)dx = 1, and
Rd
3. for any a ≤ c < d ≤ b, P (c < X < d) = c f (x)dx.
2