Download Basic mathematics – A summary

Basic mathematics – A summary
Sets
N: natural numbers 1, 2, 3, . . .
Z: integers, i.e. . . . , −2, −1, 0, 1, 2, . . .
Q: rational numbers, i.c. quotients of integers
R: real numbers, that are all numbers on the real line
Intervals
x ∈ [a, b) ⇔ a ≤ x < b
x ∈ [−∞, a) ⇔ x < a
Union: x ∈ A ∪ B ⇔ (x ∈ A or x ∈ B)
Intersection: x ∈ A ∩ B ⇔ (x ∈ A and x ∈ B)
Subset: A ⊂ B ⇔ (x ∈ A ⇒ x ∈ B)
Triangular inequality: |a + b| ≤ |a| + |b|
Cartesian coordinates
√
Distance between (x1 , y1 ) and (x2 , y2 ) is (x2 − x1 )2 + (y2 − y1 )2
y − y1
y2 − y1
The line through these points has the equation
=
x − x1
x2 − x1
Circle with center (a, b) and radius r: (x − a)2 + (y − b)2 = r2
Parabola with focus (a, b) and directrix y = p: (x − a)2 + (y − b)2 = (y − p)2 ,
b+p
1
(x − a)2 +
so y =
2(b − p)
2
x2 y 2
Ellipse: 2 + 2 = 1
a
b
x2 y 2
y 2 x2
Hyperbola: 2 − 2 = 1 or 2 − 2 = 1
a
b
b
a
Function
A function f on a set D to a set S is a rule, that maps each element x in D to an element
f (x) in S
D is the domain of the function, notation D = D(f )
The range of f is R(f ) = {f (x)|x ∈ D}
A function f is even if f (−x) = f (x) for all x ∈ D(f )
A function f is odd if f (−x) = −f (x) for all x ∈ D(f )
Let f and g be functions, where D(g) ⊂ R(f ), then the composite function g ◦ f is
given by (g ◦ f )(x) = (g(f (x)) for all x ∈ D(f )
P is a polynomial of degree n if P (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , where an ̸= 0
P (x)
, where P and Q are polynomials
R is a rational function if R(x) =
Q(x)
1
Division of rational functions
Let A be a polynomial of degree m and B a polynomial of degree n, where m ≥ n, then
A(x)
R(x)
= Q(x) +
B(x)
B(x)
where Q is a polynomial of degree m − n and R is a polynomial of degree k, where k < n
Goniometry
sin x
tan x =
cos x
sin(x + 2kπ) = sin x for k ∈ Z
cos(x + 2kπ) = cos x for k ∈ Z
degrees 0o 30o 45o
π
radials 0 π6
4
√
sin x
0( 12 )21 2
π
sin x = cos
−x
2
)
(π
−x
cos x = sin
2
sin2 x + cos2 x = 1
60o
90o
π
π
2
3
√
1
3
2
1
sin(α + β) = sin α cos β + cos α sin β
cos(α + β) = cos α cos β − sin α sin β
so
sin(2α) = 2 sin α cos α
cos(2α) = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α
Consider a triangle with edges a, b and c, and opposite angles α, β and γ, then:
sin α
sin β
sin γ
=
=
a
b
c
a2 = b2 + c2 − 2bc cos α
Limits
lim f (x) = L if for all ϵ > 0 there is a δ > 0, such that 0 < |x − a| < δ ⇒ |f (x) − L| < ϵ
x→a
Left limit: lim− f (x)
x→a
Right limit: lim+ f (x)
x→a
Squeeze theorem
If f (x) ≤ g(x) ≤ h(x) in an open interval around a and lim f (x) = lim h(x) = L, then
x→a
lim g(x) = L
x→a
sin x
=1
x→0 x
lim
( )
1
lim f (x) = lim+ f
x→∞
x→0
(x )
1
lim f (x) = lim− f
x→−∞
x→0
x
2
x→a
Continuity
f is continuous in a, if f is defined in an open interval around a and lim f (x) = f (a)
x→a
f is left continuous, if lim− f (x) = f (a)
x→a
f is right continuous, if lim+ f (x) = f (a)
x→a
f is continuous, if f is continuous for all x in D(f )
Intermediate-value theorem
If f is continuous on the interval [a, b] and s is a number between f (a) and f (b), then
there exists a c in [a, b], such that f (c) = s
Differentiation
f (x) − f (a)
x→a
x−a
f is differentiable in a, if f is defined in an open interval around a and f ′ (x) = lim
exists
f ′ (x) is the slope of the tangent to the graph in (x, f (x))
′
Product rule: (f
= f ′ (x)g(x) + f (x)g ′ (x)
(g))(x)
′
f
g(x)f ′ (x) − f (x)g ′ (x)
Quotient rule:
(x) =
g
(g(x))2
′
′
′
Chain rule: (g ◦ f ) (x) = g (f (x))f (x)
If f (x) = xr , where r ̸= 0, then f ′ (x) = rxr−1
d
dx
d
dx
sin x = cos x
cos x = − sin x
1
d
tan x =
= 1 + tan2 x
dx
2
cos x
If f is differentiable in a, then f is continuous in a
If f is defined on (a, b) and attains a maximum in c in (a, b) and f ′ (c) exists, then f ′ (c) = 0
Mean-value theorem
If f is continuous on [a, b] and differentiable on (a, b), then there is a c in (a, b), where
f ′ (c) =
f (b) − f (a)
b−a
Increasing and decreasing
f is respectively increasing or decreasing on [a, b] if for all x1 en x2 in [a, b], where x1 < x2 ,
it holds that respectively f (x1 ) < f (x2 ) or f (x1 ) > f (x2 )
Let f be defined on an interval I and J be the same interval without endpoints. If
respectively f ′ (x) > 0 or f ′ (x) < 0 for all x in J, then f is respectively increasing or
decreasing on I
3
Implicit differentiation
∂
∂
If f (x, y) = 0, where y = y(x), then
f (x, y) + y ′ f (x, y) = 0
∂x
∂y
Linear approximation
The linearization of f about a is the function L, where L(x) = f (a) + f ′ (a)(x − a)
If f is twice differentiable between x and a, then there is an s between x and a, such that
f (x) − L(x) =
f ′′ (s)
(x − a)2
2
Taylor polynomial
The Taylor polynomial of order n of f about a is the function Pn , where
Pn (x) = f (a) + f ′ (a)(x − a) +
f ′′ (a)
f (n) (x)
(x − a)2 + · · · +
(x − a)n
2
n!
If f n + 1 times differentiable between x and a, then there is an s between x and a, such
that
f (x) − L(x) =
f (n+1) (s)
(x − a)n+1
(n + 1)!
Order symbol
f (x) = O(g(x)) for x → a if |f (x)| ≤ K|g(x))| for some K and all x in an open interval
around a
Inverse function
A function f is one-to-one if for all x1 and x2 in D(f ) it holds that
f (x1 ) = f (x2 ) ⇒ x1 = x2
If f is one-to-one, then the inverse function f −1 is given by: x = f −1 (y) ⇔ y = f (x)
D(f −1 ) = R(f ) and R(f −1 ) = D(f )
1
If y = f (x), where f ′ (x) ̸= 0, then (f −1 )′ (y) = ′
f (x)
Exponential functions
1
If a > 0, then ax+y = ax ay and a−x = x and axy = (ax )y = (ay )x
a
If a > 0 and b > 0, then (ab)x = ax bx
Logarithm
If a > 0, where a ̸= 1, then y =
a
log x ⇔ x = ay
a
log(xy) = a log x + a log y
1
a
log = − a log x
x
a
log xy = y a log x
b
1
log x
a
, so a log x = x
log x = b
log a
log a
4
Natural logarithm
ln x = e log x
1
d
ln x =
dx
x
d x
x
a
=
a
ln a
dx
x2
xn
+ ··· +
+ O(xn+1 )
ex = 1 + x +
2
n!
n
x2 x3
nx
ln(1 + x) = x −
+
− · · · + (−1)
+ O(xn+1 )
2
3
n
(
x )n
lim 1 +
= ex
x→∞
n
Inverse trigonometric functions
π
π
If − ≤ x ≤ , then y = sin x ⇔ x = arcsin y
2
2
If 0 ≤ x ≤ π, then y = cos x ⇔ x = arccos y
π
π
If − < x < , then y = tan x ⇔ x = arctan y
2
2
1
d
√
arcsin
x
=
dx
1 − x2
−1
d
√
arccos
x
=
dx
1 − x2
1
d
arctan
x
=
dx
1 + x2
Summation
n
∑
n(n + 1)
i=
2
i=1
n
∑
n(n + 1)(2n + 1)
i2 =
6
i=1
n
∑
ri − 1
ri =
als r ̸= 1
r−1
i=0
∫
Mean-value theorem for integrals
b
f (x) dx = (b − a)f (c)
If f is continuous on [a, b], then there is a c in (a, b), where
a
Integral theorem
Let f be∫ continuous on an open interval I around a. Define the function F on I by
x
f (t) dt, then F is differentiable on I and F ′ (x) = f (x)
a
∫
Notation: F (x) = f (x) dx (determined up to a constant)
∫ h(x)
d
f (t) dt = f (h(x))h′ (x) − f (g(x))g ′ (x)
dx
F (x) =
g(x)
Differential equation of order n
Determine the n-times differentiable function, such that g(x, f (x), f ′ (x), . . . , f (n) (x)) = 0
5
Elementary ∫integrals
1 r+1
If r ̸= 1, then xr dx =
x
+C
r+1
∫
1
dx = ln |x| + C
x
Substitution
∫ b
∫
′
f (g(x) g (x) dx =
a
g(b)
f (u) du
g(a)
Partial integration
∫
∫ b
b
′
b
f (x)g (x) dx = f (x)g(x)|a −
f ′ (x)g(x) dx
a
a
6