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QUT MXB100 Cheat Sheet (DRAFT) by [deleted]

QUT MXB100 Exam Cheat Sheet

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Differ­ent­iation

gradient of a line
m = rise/run = (y2-y1­)/(x2 -x1)
as lim approaches 0
m = (lim h→0) f(x + h) - f(x)/h
first derivative
f'(x) = df/dx
second derivative
f''(x) = d2f/dx2
third derivative
f'''(x)=d3f/dx3
d/dx xn = nxn-1
d/dx ln(x) - 1/x
d/dx ex = ex
d/dx sin(x) = cos(x)
d/dx cos(x) = -sin(x)
 
product rule
y= uv
 
y'uv' + vu'
chain rule
y = y(u(x))
 
dy/dx = dy/du . du/dx
quotient rule
y = u/v
 
y' = u'v - uv'/v2
rewrite gradient of line: m= f(x+h) - f(x)/h

scalar product rule d/dx (ku(x)) = ku'(x) where k is a scalar
derivative of a sum: d/dx. (u(x)+­v(x)) = u'(x)+­v'(x)

Vectors

sin(𝜃) = opposi­te/­hyp­otenuse
cos(𝜃) = adjace­nt/­hyp­otenuse
tan(𝜃0= opposi­te/­adj­acent
a2+b2=c2

Matrices

C = A+B
additi­on/­sub­tra­ction
𝐵 = 𝑘𝐴
𝑘 is scalar, 𝐴 is 𝑚 . 𝑛 matrix
𝐶 = 𝐴𝐵
if A = 𝑚 . 𝑛, B = 𝑛 . 𝑘

Trig Functions

y = a sin(bx + c) + d
y = a cos(bx + c) + d
expone­ntial function
y = ex
domain
values x can assume
range
values y can assume
amplitude = a
period = 2π/b
horizontal shift = - c/b
vertical shift = d

sin(x) starts at 0, cos(x) starts at one

Expon - e = eulers's constant.

domain­/range : _ (> or <) _

Logari­thmic Differ­ent­iation

ln(ab) = ln(a) + ln(b)
ln(a/b) = ln(a) - ln(b)
ln(ab) = b x ln(a)
ln(e) = 1
eln(x) = x

Area Between Curves

∫f(x)dx - ∫g(x)dx
f(x) = upper function g(x0 = lower function
Volume of Revolution
V = π ∫ y2 dx
Integr­ating Ration Functions
f'(x) = x/x2-1
 

Integrals

∫sin(x)dx
-cos(x) + C
∫cos(x)dx
sin(x) + C
∫e^x dx
ex + C
∫1/x dx
ln(x) + C
∫xn dx
xn+1/n+1 + C
∫ln(x) dx
xln(x) - x + C
 
scalar rule
∫ ku(x) dx = k∫u(x) dx
integral of a sum
∫(u(x) + v(x))dx = ∫u(x)dx +∫v(x)dx
derivative of intergral
d/dx∫u(x)d x= u(x)
integral of derivative
∫u'(x)dx =u (x) + C
 
Integrals of Common Functions
∫sin(nx) dx
-1/n cos(nx) +C
∫cos(nx) dx
1/n sin(nx) = C
∫enx dx
1/n enx + C
∫ln(nx)dx
1/n ln(nx) + C
 
Integr­ation by Substi­tution
∫y(u(x­))u­'(x)dx
∫y(u)du
Integr­ation by Parts
∫uv' dx = uv- ∫u'v dx
∫xndx = xn+1/n+1 + C only applies when n does NOT equal -1

when n= -1, ∫1/x dx applies

Indefinite Integral: no numbers at top of bottom.

Definite Integral: solve for a number that represents the areas under the curve from x=a to x=b
no integr­ation constant in this situation

rules

 
product rule: x multiplied together in different forms eg. y = e2ex

chain rule:
inner function u(x)
outer function: y(u)

looking for function within a function eg. y=ln(s­in(x)).
let u equal the inner function

quotient: x in both the numerator and denomi­nator eg. y = exx2

remember 1/an = a-n

Functions & Algebraic Structure

y-inte­rcept: where crosses y
solve for y when x = 0
roots: where crosses x
solve for x when y = 0
linear functions
y = mx + c
quadratic functions
y = ax2 + bx + c
turning point
x = -b/2 . a
roots of quadratic
use quadratic formula
2π = 360°
radians = degrees . π/180
Function – can have only one output, y, or each unique input, x.
Relation - can have more than one output, y, for each unique input, x.

may be be more than one root for a function. roots can also be called x-inte­rcepts and zeros

linear: mx= gradie­nt/­slope C= y-inte­rcept

quadratic: pos a = 'happy face', neg a = 'sad face'
  

Explic­it/­Imp­licit

 
Explicit: dependent variable is written explicitly in terms of the indepe­ndent.
eg. y = 3x + 5

Implicit: dependent variable is not isolated to one side of equation
eg. 3x + 5 - y = 0

Explicit differ­ent­iation: when starting with implicit from that is rearra­nge­able, rearrange then do.

Implicit differ­ent­iation: relies on the chain rule. No rearra­nging required

Differ­ential Equations

First Order Separable
f(x) dx = g(y) dy
put all x to one side and y to other
 
Power & Log Rules
ab . ac = ab+c
ab/ac=ab-c
ln(ab)= bln(a)
ln(e) = 1
eln(x) = x
Decay
dN/dt = -λN
N = amount of substance, t = time and λ is decay constant
Newton's Law of Cooling
dT/dt = -k(T-Ta)
T = Temp of object, Ta is ambient temp, t is time a k is heat transfer constant
 
*Motion Problems
v = ds/dt
s = position, v = velocity, a = accele­ration, t= time
a = dv/dt
A differ­ential equation is just a mathem­atical equation that involves deriva­tives.

can have more than one solution