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MATH1722 Mathematics Foundations: Specialist | Formula Sheet

1

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 a23a21 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...