ES193: Engineering Mathematics
Standard derivatives - cosx
-sinx
Set theory - A ⊂ S
A(circle) is a subset of S(rectangle) a ∈ A implies a ∈ S
Continuous probability distribution - random distribution - expected value
E(X) = ∫xf(x) dx = μₓ [limits ∞ -∞] μₓ: expected value f(x): probability density function x: values of random variable -∞<X<∞
Scalar - definition
Have magnitude only (represented by single numbers)
Simultaneous equation - iterative methods - convergence. definition
The variables tend to a limit (the solution) as r increases.
Standard derivatives - cothx
-cosech²x
Standard derivatives - cosechx
-cosech²x.cothx
Standard derivatives - sechx
-sechx.tanhx
Standard derivatives - k (= constant)
0
Axioms & laws of Probability - equations
0≤P(A)≤1 for all A⊂S P(S) = 1 If A∩B = ∅ the P(A∪B) = P(A) + P(B) P(∅) = 0 P(A^C) = 1 - P(A) If A⊂B then P(A) ≤ P(B) P(A\B) = P(A) -P(A∩B) P(A∪B) = P(A) + P(B) - P(A∩B)
Standard integrals - Centre of mass
1. Find area by integration 2. M = Am 3. Find moment by integration if centre of mass ȳ; Find moment about x-axis Mx → m∫Area.y = m∫xdyy (find x in terms of y) [Boundaries a and b - if symmetrical a = 0 and then double] 4. Centre of mass ȳ = mx/m NB: m is a constant (variable) x̄ = moments of mass about y-axis/total mass
Ordinary differential equation - solution by direct integration
1. Integrate both sides to remove any derivatives, continue until all are gone producing a constant each time. 2. Using boundary conditions find the values of the constants. 3. Rearrange to make chosen variable the subject of the equation.
Second order ordinary differential equation - how to solve
1. Make it into the form: a[d²y/dx²] + b[dy/dx] + Cy = f(x) 2. Find complimentary function and particular integral to give general solution of the form; y = y(cf) + y(pi) 3. Determine the arbitrary constants in y(cf) using initial or boundary conditions.
First order ordinary differential equation - solution using integrating factor
1. Make it of the form: [dy/dx] + P(x)y = Q(x) Q(x) & P(x): functions of x or constants ??? How to find integrating factor 2. Integrating factor, µ(x) = e^(∫P[x]dx), substitute in P(x) value. 3. Solution obtained from; µ(x)y(x) = ∫µ(x)Q(x)dx, substitute in calculated µ(x) 4. rearrange to make y the subject.
Second order ordinary differential equation - how to find complimentary function
1. Make the equation homogeneous. (f(x) = 0) 2. Try solution y = e^(kx). Differentiate & replace dy/dx... as well; e^(kx)[ak² + bk + c] = 0 3. Since e^(kx) ≠ 0 solve ak² + bk + c = 0 to find auxiliary/characteristic equation.
Second order ordinary differential equation - auxiliary equation - b²>4ac
2 real roots k₁ & k₂ y(cf) = Ae^(k₁x) + Be^(k₂x)
Second order ordinary differential equation - auxiliary equation - b² = 4ac
2 real roots k₁=k₂ y(cf) = (A + Bx)e^(k₁x)
Random process - A^(C)
= S\A Event occurs if A does not occur
Standard integrals - ∫f'(x)[f(x)]ⁿ⁻¹dx
= [1/n]f(x)ⁿ + C
Standard integrals - ∫f'(x)e^[f(x)] dx
= e^[f(x)] + C
Standard integrals - ∫[f'(x)/f(x)]dx
= ln|f(x)| + C
Cartesian to polar coordinates - equation
?? r² = x² +y² tanθ = y/x θ = tan⁻¹(y/x)
Polar to cartesian coordinates - equation
?? x = rcosθ y = rsinθ
Sketching polar curves - equations
???
Second order ordinary differential equation - homogeneous - definition
A 2nd order linear ODE is said to be homogeneous if f(x)=0. The solution of this homogeneous equation gives the complementary function y(cf)
Function of variables - stationary points of f(x,y) rules
A = [𝛿²f/𝛿x²][𝛿²f/𝛿y²] - [𝛿²f/𝛿x𝛿y] A<0 saddle point A>0 and 𝛿²f/𝛿x²>0 minimum point A>0 and 𝛿²f/𝛿x²<0 maximum point A = 0 test fails (beyond module)
Second order ordinary differential equation -particular integral - definition
A Particular Integral is any solution of the inhomogeneous equation.
Set theory - disjoint sets
A and B have no common elements A∩B = ∅
Ordinary differential equation - linear - definition
A differential equation is linear if the dependent variable and its derivatives have degree 1, otherwise the differential equation is nonlinear. NB: non linear functions like siny make it nonlinear
Periodic function - definition & general equation & example
A function that has a definite pattern repeated at regular intervals. f(x) = f(x +L) = f(x + 2L) = f(x + 3L) = ... (complete repetition in period, L) for example: y = sin(x)
Moment of inertia - definition & general equations
A measure of how difficult it is to rotate the lamina. (equivalent of mass in linear motion) C = Iθ̈ C: couple/torque θ̈: angular acceleration I = ∫r²dm m: mass r: distance from O to location of centre of mass of the strip. [Limits chosen so that the entire lamina is included] Units: kgm²
Function - definition
A rule which operates on an input and produces a single output from that input. A function always represents a mapping but a mapping is not necessarily a function.
Mapping - definition
A rule which operates where multiple outputs are produced. (one-to-many). A function always represents a mapping but a mapping is not necessarily a function.
Domain - definition
A set of X values used as input.
Range - definition
A set of values that y takes as x is varied.
Asymptote - definition
A straight-line which is tangent to a curve at infinity. Find by checking lim(y) as x→∞ or lim(x) as y→∞ and then rearrange for y or x respectively to find an approximate equation.
Matrix multiplication - rules
A(B + C) = AB + AC A(BC) = (AB)C AI = IA = A Aⁿ = AAA..A Iⁿ = I (AB) transpose = B^(T)A^(T) Order changes
Properties of determinant - Δ = 0
All elements in any row or column = 0 OR Two rows or columns are identical
Laplace transformation - definition
An integral transform: transforms t → s OR f(t) → F(s) L{f(t)}(s) = F(s) = ∫e^(-st) f(t) dt [boundary ∞ 0] The transform only exists when the improper integral exists (i.e. is finite)
Ordinary differential equation - definition
An ordinary differential equation for the function y(x) (or y(t)) is an equation relating x and y (or t and y) and one or more derivatives of y with respect to x (or t )
Standard integrals - ∫x/[a + x] dx
Any fraction where the top power of x is greater or equal to the bottom, use LONG DIVISION.
Simultaneous equation - iterative methods - oscillation definition
Approximations oscillate around the solution.
Second order ordinary differential equation - [d²y/dx²] - a²y = f(x) - y(cf) & auxiliary equation
Auxiliary equation: k² - a² = 0 ∴k=±a y(cf) = Ae^(ax) + Be^(-ax)
Standard integrals - Average value - equation
Av. Value = [1/(b - a)] ∫f(t) dt [limits b a]
Laplace transformation - rules
Can remove constants Performs on individual functions. (you can separate functions) s+a → multiply by e^(-at)
Parametric differentiation - how to/equation
Chain rule: dy/dx = (dy/dt)/(dx/dt) = dy/dt x (dx/dt)⁻¹ d²y/dx² = (dy/dx).(d/dx) = {1/[dx/dt]}.{d/dt[dy/dt x (dx/dt)⁻¹]} as d/dx = {1/[dx/dt]}.{d/dt}
Second order ordinary differential equation - auxiliary equation - b²<4ac
Complex conjugate pair y(cf) = A₁e^(α + iβ) + B₁e^([α + iβ]x) = e^(αx)[Acos(βx) + Bsin(βx)] A = A₁ + B₁ B = A₁ - B₁
Set theory - A^(C)
Compliment of A Set of elements not belonging to A A^(C) = { x·x∉A and x ∈S} = S\A where A⊂S
Standard derivatives - rules
Constants can be removed Individual variables can be differentiated separately Product rule Quotient rule Chain rule
Continuous probability distribution - random variable
Continuous random variable X is a function that assigns a real number to each event A⊂S X:S →~ X is a "measure related to the random process - ∞ < X < ∞ P(a≤X≤b): probability of a≤X≤b occurring eg amount of time before computer crashes & diameter of ball bearings produced in manufacturing process
Standard integrals - improper integrals - definition
Definite integrals where the integrand becomes infinite at some point in the interval of integration OR Where the length of the interval of integration is itself infinite e.g ∫dx/√(9 - x²) in the range 0 to 3
Simultaneous equation - iterative methods - divergence definition
Difference between xᵤⁿ⁺¹ and xᵤⁿ grows with increasing r (= n in this eg). This does not necessarily mean that a solution does not exist.
Set theory - A\B
Difference of A and B set of elements belonging to A but not B A\B = {x·x∈A and x∉B}
Discrete probability distribution - random variable
Discrete random variable X is a function that assigns a real number to each outcome s∈S X:S →~ X is a "measure related to the random process P(X=a): probability of X=a occurring
Discrete probability distribution - random distribution -expected value - equation
E(X) = ∑xif(xi) = µₓ [i is subscript & limits n i = 1] Mean of X f(xi): probability distribution function xi: value of random variable X
Periodic function - cycle - definition
Each compete pattern.
Set theory - ∅
Empty set ∅ ⊂ S for all sets S
Independent events - definition & equation
Event A is independent of event B (use equation below to confirm if events are independent) P(A∩B) = P(A)P(B)
Random process - A∩B
Event occurs if A and B occur
Random process - A∪B
Event occurs if A or B occurs
Random process - A∩B = ∅
Events A and B are mutually exclusive.
Ordinary differential equation - solution by separation of variables
FORM: [dy/dx] = f(x)g(y) 1. Divide through by g(y). 2. Integrate both sides: ∫[1/g(y)] dy] = ∫f(x) dx 3. Complete integration, using boundary conditions find the values of the constants. 4. Rearrange to make chosen variable the subject of the equation.
Moment of inertia - rod and lamina equation
Finding moment of inertia about the y-axis mass = ml𝛿x l𝛿x: Area of strip (rectangular strip height l.𝛿x width) m: mass per unit area I = ∑mlx²𝛿x = ∫mlx²dx [with limits to include the whole area - I about each axis] AT END sub in m = M/area M: mass m: specific mass (for rod use was per unit length and remove l from the equation)
Moment of inertia - parallel axis theorem & equation
For axis L', parallel to L and both lying in the same plane as the lamina (mass M) where L passes through centre of mass of lamina. I(L') = I(L) + Ml² I = length between axis L' and L (not an i but an l) Things in brackets are subscript
Standard integrals - substitution indefinite vs definite
For definite integrals you need to change the limits. For indefinite & definite you must change to du.
Continuous probability distribution - random distribution - cumulative distribution function rules & equation
Fₓ(a) = P(X≤a) = ∫f(x) dx -∞<a<∞ [limits a -∞] Can be written like probability density or in table if a≤ b Fₓ(a) ≤ Fₓ(b) lim{x →-∞} Fₓ(x) = 0 lim{x →∞} Fₓ(x) = 1 [d/dx]Fₓ(x) = f(x)
Discrete probability distribution - cumulative distribution function - equation & rules
Fₓ(xi) = P(X≤xi) = ∑f(xj) if xi ≤ xj then Fₓ(xi) ≤ Fₓ(xj) Fₓ(x₁) = f(x₁) Fₓ(xn) = 1 Displayed as a table, ...
Standard integrals - definite integration
Gives the area to the x-axis. (a number not a function is given)
Set theory - when does A = B
If A ⊂B and B ⊂ A
Properties of determinant - Δ no change
If all elements from one row or column are scaled and added to another row or column.
Properties of determinant - Δ = λΔ
If all members in a row or column are scaled by a factor λ
Moment of inertia - perpendicular axis theorem & equation
If lamina lies in x-y plane; I(OZ) = I (OX) + I(OY) I: moment of inertia - subscription shows about which axis Things in brackets are subscript
Properties of determinant - Δ sign change
If two rows or columns are interchanged.
Set theory - A∩B
Intersection (only) of A and B Set of element belonging to A and B A∩B = {x·x∈A and x∈B}
Confidence interval - definition
Interval within which we would expect to find an estimate of a parameter (at specified probability level)
Laplace transformation - pairs
L & L⁻¹ (inverse) form a Laplace transformation pair f(t) = L⁻¹{F(s)} You may need to rearrange/use partial fractions to find L⁻¹ or complete the square to give form from tables.
Standard integrals - length of a curve - equation
Length of arc = S = ∫√[1+(dy/dx)²] dx Formula given If in parametric form: S = ∫√[(dx/dt)² + (dy/dt)²] dt NB: change limits and dx to dt
Laplace transformation - derivatives
L{(dy/dt)(t)} = ∫e^(-st) (dy/dt)(t) dt [boundary ∞ 0 & solve using by parts] Turn from differential → algebraic L{(dy/dt)(t)} = sY(s) - y(0) L{(d²y/dt²)(t)} = s²Y(s) - sy(0) - y'(0) y(0): found with conditions y'(0): found with conditions Rearrange for Y(s) to be the subject of the equation You can find the inverse the same way
Inverse matrix - rules
Must be a square matrix detA = |A| = Δ ≠ 0 if Δ = 0 A is a singular matrix and A⁻¹ doesn't exist A⁻¹A = AA⁻¹ = I A is inverse of A⁻¹ & vice versa
Continuous probability distribution - standardised normal distribution -
N(μ,σ²) Standardised: U = [X - μ]/σ P(X≤b) = P(U≤β) b also needs to be transformed: β = [b -μ]/σ E(U) = 0 Var(U) = 1 ∴ N(0,1) Symmetric about y axis
Normal → poisson distribution approximation - equation & conditions
N(μ,σ²) approximation for p(k;λ) μ = λ σ² = λ - λ is large
Exponential & Poisson distribution relationship
Number of events per interval of time - Poisson distribution Length of time between events - Exponentially distributed ???
Conditional probability - definition & equations
P(A|B) The probability of event A occurring given that the event B has occurred P(A|B) = P(A∩B)/P(B) P(B) = P(B|A)P(A) + P(B|A^C)P(A^C) P(A∪B|C) = P(A|C) + P(B|C) - P(A∩B|C)
Bayes' Theorem - equation
P(A|B) = [P(B|A)P(A)]/P(B) for S = A₁∪A₂∪...∪An Ai∩Aj = ∅ P(Ai|B) = P(Ai)P(B|Ai)/[P(A₁)P(B|A₁) + ... + P(An)P(B|An)]
Standard integrals - ∫f(x)/[ax + b]²
Partial fractions: [A/(ax + b)] + [B/(ax + b)²]
Standard integrals - ∫f₁(x)/f₂(x)[ax² + bx + c]
Partial fractions: [A/f₁(x)] + [(Bx + C)/(ax² + bx + c)] When ax² + bx + c will not factorise into simple linear factors
Second order ordinary differential equation -particular integral - common f(x) equations
Polynomials, exponentials, sines and cosines, sinh & cosh
Random process - P(A)
Probability Relative frequency of event A⊂S occurring in a random process sample space S
Discrete probability distribution - poisson distribution - conditions
Random process Repeated occurrence of single event in fixed interval
Discrete probability distribution - binomial distribution - conditions
Random processes Repeated independent events Each event has two possible outcomes; success or failure P(success) = p P(failure) = q = 1 - p
Standard derivatives - y = logₐx
Rearrange for x = a^(y), ln (logₑ) both sides then rearrange for y dy/dx = [1/logₑa] . [1/x]
Inverse function - how to find equation
Rearrange to have x =... then replace x with y?? Only one-to-one functions have inverses.
Transpose matrix - how to make and rules
Reflect along leading diagonal of square formed with upper left corner. (n = T) (Aⁿ)ⁿ = A If: Aⁿ = A matrix is symmetric Aⁿ = -A matrix is skew symmetric
Simultaneous equation - iterative methods - advantages
Requires fewer operations
Random process - definitions
Results in varying outcomes (events) Every outcome is unpredictable Sample point s∈S (element of S) Event A⊂S (set of actual outcomes of random process)
Random process - S
Sample space S certain or sure event
Poisson → binomial distribution approximation - equation & conditions
Set λ = np p(k;λ) gives approximation for b(k;n,p) Valid for: n - large p - small np - moderate size
Discrete probability distribution - standardised random variable - equation & rules
Standardised random variable = X* = [X - μₓ]/σₓ μₓ: expected value σₓ: standard deviation E(X*) = 0 Var(X*) = 1
Standard integrals - ∫N(x)/D(x) dx where both are functions involve sums of only sinx and cosx
Substitutions: sinx = 2t/(1 + t²) cosx = (1 - t²)/(1 + t²) dx = 2dt/(1 + t²) t = tan(x/2)
Standard integrals - ∫N(x)/D(x) dx where both are functions involve even powers of sinx and cosx only
Substitutions: sin²x = T²/(1 + T²) cos²x = 1/(1 + T²) dx = dT/(1 + T²) t = tanx
Standard derivatives - y = a^(x)
Take logs both sides and then rearrange for x dy/dx = ylna = a^(x)lna
Ordinary differential equation - degree - definition
The degree of a differential equation is the power to which the highest derivative is raised.
Argument - definition
The input of the function.
Periodic function - period - definition
The interval over which the repetition takes place.
Inverse function - how to draw graphically
The inverse function f⁻¹(x) of y = f(x) is the reflection in the line y = x.
Modulus of inequalities - equation
The modulus defines the magnitude. |2x - 1|≥3 so 2x - 1 ≥ 3 OR 2x - 1 ≤ -3
Ordinary differential equation - order - definition
The order of a differential equation is the order of its highest derivative.
Derivatives - definition & symbols
The rate of change of y with respect to x (tangent to the curve). d: infinitésimal change 𝛿: finite change
Ordinary differential equation - general solution - definition
There are many different solutions which can satisfy a particular differential equation. A solution from which all possible solutions can be found is called a general solution.
Standard derivatives - function of n variables
There are n gradients where you can find the partial (δ) derivative For this treat every variable but the one you are differentiating with respect to as a constant
Total differential - cantor plots - definition
They show the location of points that all have the same value e.g when f(x,y) = c (constant). Along these lines the total differential has to be zero.
Set theory - A∪B
Union of A and B Set of elements belonging to A or B (intersection & circles) A∪B = {x·x∈A or x∈B}
Normal - equation
Use three points on a plane N⃗ = (c⃗-a⃗)x(b⃗-a⃗) ^cross product gives normal to the two vectors (CA and BA) in the plane where c⃗, a⃗ and b⃗ are point vectors from the origin
Standard integrals - ∫sinAxsinBx dx, ∫sinAxcosBx dx OR ∫cosAxcosBx dx
Use trigonometric functions like; 2sinAcosB = sin(A + B) + sin(A - B)
Discrete probability distribution - random distribution - variance - equation
Var(X) = ∑(xi - μₓ)²f(xi) = E(X²) - μₓ² = σ² [i is subscript & limits n i = 1] μₓ: expected value f(xi): probability distribution function xi: value of random variable X σₓ: standard deviation
Continuous probability distribution - random distribution - variance - equation
Var(X) = ∫(x - μₓ)²f(x) dx = σₓ² [limits ∞ -∞] = E(X²) - μₓ² μₓ: expected value f(x): probability density function x: values of random variable -∞<X<∞ σₓ: standard deviation
Vector - definition & rules
Vectors have magnitude and direction they are not fixed in space but can be translated. AB ≠ BA BA = -AB b̲ = a̲ if both magnitude (modulus) and direction (θ) are the same
Continuous probability distribution - standardised random variables - equation & rules
X* = [X - μₓ]/σₓ σₓ: standard deviation μₓ: Expected value X: random variable E(X*) = 0 Var(X*) = 1
Standard integrals - indefinite rules
You can remove constants You can operate individual functions (if addition/... but not if multiplication/...)
Standard integrals - definite integration rules
You can remove constants You can operate individual functions (if addition/... but not if multiplication/...) For even functions where f(x) = f(-x) (symmetrical) ∫f(x)dx for the whole range = 2∫f(x)dx for 0 + (half the range) For odd functions where f(x ) = -f(-x) (rotational symmetry) ∫f(x)dx for whole range = 0 Average value of function over range a≤x≤b is 1/[b-a]∫f(x)dx for a to b range. Integration by parts Product rule Quotient rule Chain rule
Standard integrals - xⁿ
[1/n+1]xⁿ⁺¹ + c when n≠-1
Standard integrals - indefinite integration
[dF(x)/dx] = f(x) F(x) is the anti-derivative of f(x). You must add constant C so the antiderivative is not unique.
Acceleration - differential equation
a = dv/dt = d²s/dt²
Complete the square - general equation
ax² + bx + c = 0 → a(x+[b/2a])² + (c − [b²/4a]) = 0
Dot product - equation & rules
a⃗.b⃗ = |a⃗||b⃗|cosθ (scalar & you can rearrange for cosθ) a⃗.b⃗ = 0 cosθ = 0 so a⃗ and b⃗ are perpendicular a⃗.a⃗ = |a⃗|² as cosθ =1 (a b c).(d e f) = ad + be + cf [d/dt](a.b) = da/dt . b + a . db/dt [d/dt](a.a) = 2a . da/dt
Cross product - definition & rules
a⃗xb⃗ = |a⃗|.|b⃗|sinθn̲̂ (vector & found like determinant but with i,j,k as top row) n̲̂: = n/|n| unit vector perpendicular to a⃗ and b⃗in the direction which a right hand screw would move if a⃗ was rotated towards b⃗. a⃗xb⃗ ≠ b⃗xa⃗ axb = 0 a||b [d/dt](axb) = da/dt x b + a x db/dt axb is perpendicular to both a and b
Discrete probability distribution - binomial distribution - probability of success - equation
b(k;n,p) (n over k) p^(k) q(n-k) k = 0,1,...,n (n over k): n!/k!(n - k)! k: number of successes n: number of events p: P(success) q: P(failure) = 1 - p Probability of no successes b(0;n,p) = q^n Probability of ≥1 successes = 1 - q^n
Standard derivatives - sinhx
coshx
Hyperbolic functions - equations
cosh²x - sinh²x = 1 (convert into exponentials to prove)
Standard derivatives - sinx
cosx
Angle between two vectors - equations
cosθ = (a⃗.b⃗)/(|a⃗||b⃗|) |sinθ| = |a⃗xb⃗|/(|a⃗||b⃗|)
Standard derivatives - ln(f[x])
dy/dx = [1/f(x)] . f'(x)
Total differential - equation
dz = [𝛿f/𝛿x]dx + [𝛿f/𝛿y]dy
Newton's Law of cooling - differential equation
dθ/dt = -kθ Body radiating heat
Random process - {a} ⊂ S
elementary event??
Odd function - definition & equation
f(x) = -f(-x) The graph possesses rotational symmetry about the origin (its the same if rotated 180°).
Even function - definition & equation
f(x) = f(-x) The graph is symmetrical about the vertical axis.
Discrete probability distribution - probability distribution function - equation & rules
f(xi) = P(X = xi), I = 1,2,...,n f(xi)≥0 ∑f(xi) = 1 Displayed as a table showing probability of each event occurring, ...
Continuous probability distribution - random distribution - probability density function & rules
f(·): probability density function Written as f(x) { list of functions and conditions where functions are applicable eg 0, elsewhere f(x) ≥ 0 ∫f(x) dx = 1 [limits ∞ -∞] p(a≤X≤b) = ∫f(x) dx [limits b a]
Normal → binomial distribution approximation - equation & conditions
for binomial distribution b(k;n,p) from normal distribution N(μ,σ²) μ = np σ² = npq Provided: - n is large - neither p nor q are close to zero
i - equation & powers
i = √-1 even powers are real numbers
Random process - ∅
imposible event
Line of intersection of two planes - equation
line of intersection direction = b⃗ = N⃗₁ x N⃗₂ equation of line of intersection: r⃗ = a⃗ + λb⃗ = a⃗ + λ(N⃗₁ x N⃗₂) where a is a vector to any point on that line
Logarithmic function - rules
logₐ1 = 0 logₐa = 1 logₐ(x₁x₂) = logₐx₁ + logₐx₂ logₐ(x₁/x₂) = logₐx₁ - logₐx₂ logₐxⁿ = nlogₐx logₒx = logₐx/logₐo n = lnx x = eⁿ
Gradient - equation
m = vertical distance/horizontal distance
Hyperbolic functions - coshx exponential equation & inverse
n = x coshx = [eⁿ + e⁻ⁿ]/2 sechx = 1/coshx
Hyperbolic functions - sinhx exponential equation & inverse
n = x sinhx = [eⁿ - e⁻ⁿ]/2 cosechx = 1/sinhx
Standard derivatives - xⁿ
nxⁿ⁻¹
Discrete probability distribution - poisson distribution - probability of k occurrences - equation
p(k;λ) = [(λ^k)/k!] e^(-λ), k = 0,1,2... k: total number of events n: number of units in the data λ: k/n
Standard integrals - root-mean-square - equation
r.m.s value = √{[1/(b - a)] ∫[f(t)]² dt} [limits b a] For any sinusoidal waveform over an interval equal to one period r.m.s value = [1/√2](0.707) x the amplitude of the waveforrm
Set theory - s ∈ S
s is an element of S
Standard derivatives - tanhx
sech²x
Standard derivatives - tanx
sec²x
Standard derivatives - coshx
sinhx
Hyperbolic functions - tanhx exponential equation & inverse
tanhx = sinhx/coshx cothx = 1/tanhx
Standard integrals - surface of revolution - equation
total surface area = ∫2πy√[1 + (dy/dx)²] dx
Standard integrals - ∫sin²x dx OR ∫cos²x dx
use cos2x = cos²x - sin²x = 1 - 2sin²x = 2cos²x -1
Standard integrals - ∫sinⁿxcosⁱx dx where i = m
use a substitution u = sinx OR u = cosx
Velocity - differential equation
v = ds/dt
Standard integrals - common substitutions √(x² - a²)
x = acoshu OR x = asecu
Standard integrals - common substitutions √(x² + a²)
x = asinhu OR x = atanu
Standard integrals - common substitutions √(a² - x²)
x = asinu OR x = acosu
Standard integrals - common substitutions x² + a²
x = atanu
Ordinary differential equation - [dy/dx] + y = x² - dependent variable
x is the dependent variable DEF???
Quadratic - equation & parameters
x=(-b±√[b²-4ac])/2a b² = 4ac : one solution exists b² > 4ac : two solutions exist b² < 4ac : no (real) solutions
Standard derivatives - inverse trigonometric functions
y = sin⁻¹x sine both sides to get siny = x and then use implicit differentiation OR use d/dy (instead of d/dx) and then inverse Use trigonometric functions to rewrite in terms of x.
Ordinary differential equation - [dy/dx] + y = x² - independent variable
y is the independent variable DEF???
Second order ordinary differential equation - [d²y/dx²] + a²y = f(x) - y(cf) & auxiliary equation
y(cf) = Acos(ax) + Bsin(ax)
Complex numbers - converting into modulus argument form
z = x + iy modulus = r = √(x² + y²) argument = θ = tan⁻¹(y/x) -π ≤ θ ≤ π z = r(cosθ + isinθ) OR z = re^(iθ) as e^(iθ) = cosθ + isinθ so x = rcosθ y = rsinθ
Complex numbers - nth root - equation
z^(1/n) = r^(1/n) (cos[{θ + k2π}/n] + isin[{θ + k2π}/n]) substitute in n values of k = -1, 0, 1, 2, ... plotted answers should be an equal distance 360° around the Argand diagram
Simultaneous equation - iterative methods - conditions to stop iteration
|xᵣⁿ⁺¹-xᵣⁿ| < ε ε = 10⁻⁴
Discrete probability distribution - binomial distribution - expected value - equation
μ = np k: number of successes n: number of events p: P(success) q: P(failure) = 1 - p
Discrete probability distribution - poisson distribution - expected value - equation
μ = λ = mean k: total number of events n: number of units in the data λ: k/n
Discrete probability distribution - binomial distribution - variance - standard deviation - equation
σ = √(npq) k: number of successes n: number of events p: P(success) q: P(failure) = 1 - p
Discrete probability distribution - poisson distribution - standard deviation - equation
σ = √λ k: total number of events n: number of units in the data λ: k/n
Discrete probability distribution - binomial distribution - variance - equation
σ² = npq k: number of successes n: number of events p: P(success) q: P(failure) = 1 - p
Discrete probability distribution - poisson distribution - variance equation
σ² = λ k: total number of events n: number of units in the data λ: k/n
Discrete probability distribution - random distribution - standard deviation
σₓ = √[Var(X)] Var(X): Variance
Continuous probability distribution - random distribution - standard deviation
σₓ = √[Var(X)] σₓ: standard deviation
Standard integrals - trigonometric forms
∫N(x)/√[ax² + bx + c] dx or without √ Complete the square to make the form given in the formula booklet.
Standard integrals - volume of revolution about the y-axis
∫πx²dy [boundaries d & c, leave in terms of π...]
Standard integrals - volume of revolution about x-axis
∫πy²dx [boundaries b & a, leave in terms of π...]