Title | MATH1722 Formula Sheet |
---|---|
Course | Mathematics Foundations: Specialist |
Institution | University of Western Australia |
Pages | 8 |
File Size | 247.5 KB |
File Type | |
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MATH1722 Mathematics Foundations: Specialist | Formula Sheet
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School of Mathematics and Statistics
MATH1722 MATHEMATICS FOUNDATIONS: SPECIALIST FORMULA SHEET Algebra a + b = b + a , ab = ba (a + b) + c = a + (b + c) , (ab)c = a(bc) a(b + c) = ab + ac
Commutative law Associative law Distributive law
To expand (a + b)n for any positive integer n, the powers are an , an−1 b, an−2 b2 and so on until a2 bn−2 , abn−1 and bn . Pascal’s triangle can be used to find the coefficients.
1 1 1 1 2 1 1 3 3 1
Absolute values The absolute value |a| is defined as |a| = Properties: |a| = | − a| =
√
a , a≥0 −a , a < 0
a2 > 0
For b > 0:
|ab| = |a||b| a |a| = |b| b
|a| = b iff a = ±b |a| < b iff − b < a < b
|an | = |a|n
|a| > b iff a < −b or a > b
Inequalities If a < b and b < c then a < c
If 0 < a < b then
1 1 > a b
If a < b then a + c < b + c
If a < b and c > 0 then ac < bc
If a < b and c < d then a + c < b + d
If a < b and c < 0 then ac > bc
Pythagoras’ theorem
a
c
b
a2 = b2 + c2.
MATH1722 Mathematics Foundations: Specialist | Formula Sheet
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Linear equations Slope-intercept form: y = mx + b. Slope = m = Slope-point form: y − y1 = m(x − x1 ).
y2 − y1 . x2 − x1
Quadratic equations General quadratic equation ax2 + bx + c has roots √ −b ± b2 − 4ac x= , 2a where b2 − 4ac is the discriminant. Functions Even function: f (−x) = f (x); Odd function: f (−x) = −f (x). One-sided limits: x → c− , x → c+ ; Two-sided limit: x → c where
lim f (x) = L iff lim f (x) = L and lim f (x) = L. − + x→c
x→c
x→c
L’Hospitals rule: If lim f (x) = 0 = lim g(x) or lim f (x) = ±∞ = lim g(x) so that x→a
indeterminate form of type
x→a
0 0
or
∞ , ∞
x→a
x→a
f g
is an
then
f ′ (x) f (x) . = lim ′ x→a g (x) x→a g(x) lim
Tangent to a curve y = f (x) at x = a has slope f ′ (a). The function y = f (x) is increasing/decreasing in the interval (a, b) provided that f ′ (x) > 0/f ′ (x) < 0 for all values of x in (a, b). The function y = f (x) is concave up/concave down in the interval (a, b) provided that f ′′(x) > 0/f ′′(x) < 0 for all values of x in (a, b). The function y = f (x) has a critical point/inflection point at x = a if f ′ (a) = 0/f ′′(a) = 0. The first derivative test: For a critical point c of f : • If f ′ (x) > 0 to the left of c and f ′ (x) < 0 to the right of c, then f (c) is a local maximum of f . • If f ′ (x) < 0 to the left of c and f ′ (x) > 0 to the right of c, then f (c) is a local minimum of f . • If f ′ (x) > 0 to the left and right of c, or f ′ (x) < 0 to the left and right of c, then f (c) is neither a local maximum or local minimum of f . The second derivative test: For a critical point c of f : • If f ′′(c) < 0 then f (c) is a local maximum of f . • If f ′′(c) > 0 then f (c) is a local minimum of f .
MATH1722 Mathematics Foundations: Specialist | Formula Sheet
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Exponential and logarithmic functions Exponential function: y = ax
y
y = ax
Domain R , Range (0, ∞) lim ax = ∞ ,
y=x
lim ax = 0
x→∞
x→−∞
Natural exponential function: y = ex
y = loga x x
Logarithmic function: y = loga x Domain (0, ∞) lim loga x = ∞ ,
x→∞
, Range R lim loga x = −∞
x→0+
Natural logarithmic function: y = loge x = ln x Cancellation equations: loga(ax ) = x and alog a x = x. Index laws: a0 = 1 1
a2 =
√
ax = ax−y ay
ax ay = ax+y a−x =
a
1 ax
(ab)x = ax bx
(ax )y = axy a x b
=
ax bx
Log laws: loga 1 = 0
loga a = 1
y
loga(x ) = y loga x
loga(xy) = loga x + loga y
x = loga x − loga y loga y
loga x =
logc x logc a
Trigonometry
H
θ
O
A
O sin θ = H ; cos θ = HA ; tan θ = OA = O/H A/H
=
sin θ cos θ .
cosec θ = sec θ = cot θ =
1 sin θ
1 cos θ 1 tan θ
=
H O;
=H A; =
A O
MATH1722 Mathematics Foundations: Specialist | Formula Sheet
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Reference triangles for common angles:
√
π 3
π 4
2
2
1 π 4
π 6
1
1
√
3
Trigonometric functions:
y = sin x Domain R Range [−1, 1]
y = cos x Domain R Range [−1, 1]
y = tan x Domain x 6= π2 + nπ Range R
y = cosec x Domain x 6= nπ Range R
y = sec x Domain x 6= π2 + nπ Range R
y = cot x Domain x 6= nπ Range R
MATH1722 Mathematics Foundations: Specialist | Formula Sheet Trigonometric properties: Fundamental property: sin2 x + cos2 x = 1, tan2 x + 1 = sec2 x, 1 + cot2 x = cosec2 x. Odd/even properties: sin(−x) = − sin x, cos(−x) = cos x. Addition formula: sin(x + y) = sin x cos y + cos x sin y, cos(x + y) = cos x cos y − sin x sin y, tan(x + y) =
tan x + tan y . 1 − tan x tan y
Half-angle formula: sin(2x) = 2 sin x cos x,
cos(2x) = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x, tan(2x) =
2 tan x . 1 − tan2 x
Product formula: sin x cos y = 21 [sin(x + y) + sin(x − y )],
sin x sin y = 21 [cos(x − y) − cos(x + y )],
cos x cos y = 21 [cos(x + y) + cos(x − y )]. Differentiation The product rule: If y = uv then
du dv dy =v +u . dx dx dx
dv du v −u dy u = dx 2 dx . The quotient rule: If y = then v dx v The chain rule: If y = f (u) and u = g(x) then
du dy dy = × . dx du dx
Integration Zb
f (x) dx = −
a
Za
f (x) dx
Zb
f (x) dx where c is a constant,
,
b
Zb
cf(x) dx = c
Zb
Zb
a
f (x) dx = 0,
a
a
f (x) dx +
g(x) dx =
Zb
a
a
a
Zb
Zc
Zb
a
Za
f (x) dx =
a
f (x) dx +
c
[f (x) + g (x)] dx,
f (x) dx where a < c < b.
5
MATH1722 Mathematics Foundations: Specialist | Formula Sheet
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The fundamental theorem of calculus: If the function f is continuous on the interval [a, b] and if F ′ (x) = f (x), then for a definite integral Zb f (x) dx = F (b) − F (a) = [F (x)]ab, a
while for an indefinite integral Z
f (x) dx = F (x) + C,
where C is an arbitrary integration constant.
Area between curves:
Zb
[ytop (x) − ybot(x)] dx or
a
Zb
[xright(y) − xleft(y)] dy .
a
1 Average value of a function: y¯ = b−a
Zb
f (x) dx.
a
Matrices
A = [aij ]m×n
a11 a21 = .. .
a12 a22 .. .
am1 am2
· · · a1n · · · a2n . .. . . . · · · amn
Equality: Two m × n matrices A = [aij ]m×n and B = [bij ]m×n are equal iff aij = bij for all i and j . Addition/subtraction: If A = [aij ]m×n and B = [bij ]m×n are m × n matrices then A ± B = [aij ± bij ]m×n . Scalar multiplication: If k is any real number then the scalar multiple of the matrix A = [aij ]m×n is kA = [kaij ]m×n . Matrix multiplication: For a matrices A = [aij ]m×p and B = [bij ]p×n the product AB is an m × n matrix whose entry in the ith row and j th column is the dot product of a vector given by the ith row of A with a vector given by the j th column of B . To calculate det A = |A| for 2 × 2 or 3 × 3 matrices:
−
a11 a12 a21 a22
+
a11 a21 a31 − −
a12 a22 a32 −
a13 a11 a23a21 a33 a31 +
a12 a22 a32 + +
MATH1722 Mathematics Foundations: Specialist | Formula Sheet
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A system of n linear equations in n unknowns may be written as a marix equation Ax = b, where a11 a12 · · · a1n x1 b1 a21 a22 · · · a2n x2 b2 .. . , x = .. , b = .. , A = .. . . . . . . . . . an1 an2 · · · ann xn bn
and A is the coefficient matrix.
The system has a unique solution iff det A 6= 0. 3D vectors v = (v1 , v2 , v3 ) = v1 i + v2 j + v3 k ,
i = (1, 0, 0) ,
j = (0, 1, 0) ,
k = (0, 0, 1).
Addition, subtraction, scalar multiplication: u ± v = (u1 , u2 , u3 ) ± (v1 , v2 , v3 ) = (u1 ± v1 , u2 ± v2 , u3 ± v3 )
= (u1 i + u2 j + u3 k) ± (v1 i + v2 j + v3 k) = (u1 ± v1 )i + (u2 ± v2 )j + (u3 ± v3 )k,
kv = k(v1 , v2 , v3 ) = (kv1 , kv2 , kv3 ) = k(v1 i + v2 j + v3 k) = kv1 i + kv2 j + kv3 k. Norm of a vector v = (v1 , v2 , v3 ) : |v| = Unit vector parallel to a vector v : v ˆ=
q
v12 + v 22 + v 23 =
1 v. |v|
√
v · v.
Dot product of two vectors u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ): u · v = u1 v1 + u2 v2 + u3 v3 = |u||v| cos θ.
a·b . The component of a along b : compba = a·bˆ = |b|
Cross product of two vectors u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ): i j k u × v = u1 u2 u3 = (u2 v3 − u3 v2 )i + (u3 v1 − u1 v3 )j + (u1 v2 − u2 v1 )k. v 1 v 2 v 3
The norm of the cross product: |a × b| = |a||b| sin θ .
Triple scalar product and triple vector product of three vectors a, b and c: a1 a2 a3 a · (b × c) = b1 b2 b3 , a×(b × c) = (a · c)b − (a · b)c. c 1 c 2 c 3
Parametric equation of a line L connecting the points defined by the heads of two vectors a and b: r(t) = a + t(b − a) where r(0) = a and r(1) = b.
MATH1722 Mathematics Foundations: Specialist | Formula Sheet
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Differentiation and integration formulas
dy dx
y
0
a (constant)
ax + C
nxn−1
xn (n 6= −1)
xn+1 +C n+1
1 or − x−2 x2
1 or x−1 x
ln x + C
ax ln a
ax
ax +C ln a
ex
ex
ex + C
1 x ln a
loga x
1 (x ln x − x) + C ln a
1 x
ln x
x ln x − x + C
cos x
sin x
− cos x + C
− sin x
cos x
sin x + C
sec2 x
tan x
ln(sec x) + C
− cot x cosec x
cosec x
ln(cosec x − cot x) + C
tan x sec x
sec x
ln(sec x + tan x) + C
−cosec2 x
cot x
ln(sin x) + C
−
Z
y dx...