| Sets: |
| ℕ0 = Natural Numbers: {0,1,2,3,...} |
| ℕ1 = Natural Numbers: {1,2,3,...} |
| ℤ = Integers: {...,-3,-2,-1,0,1,2,3,...} |
| ℚ = Rational Numbers: any value expressed as a ratio a/b where a and b are any integers |
| ℝ = Real Numbers: any value that represents a quantity along a continuous line |
| ℂ = Complex Numbers: any quantity in the form a+bi where a and b are real numbers and i is the imaginary constant |
| ∅ = the Empty Set with no elements: {} |
| U = the Universal Set (all possible elements) |
| Set Operators: |
| A ∈ B → A is an element of set B |
| A ∉ B → A is not an element of set B |
| A ∪ B → union of sets A and B (all elments of both) |
| A ∩ B → intersection of sets A and B (only elments in common) |
| A ⊂ B → A is a proper subset of B (A ≠ B) |
| A ⊆ B → A is a subset of B |
| 2A → power set of A (set of all subsets) |
| Constants: |
| π = 3.141592653589793... ratio of circumference to diameter of circle |
| e = 2.718281828459045... Euler's number |
| i = the imaginary constant |
| φ = 1.61803398874989... golden ratio |
| Operator Precedence: |
|
| Relational Symbols: |
| x = y → x is equal to y |
| x ≠ y → x is not equal to y |
| x <> y → x is not equal to y |
| x ≈ y → x is approximately equal to y |
| x > y → x is greater than y |
| x ≫ y → x is much greater than y |
| x ≥ y → x is greater than or equal to y |
| x < y → x is less than y |
| x ≪ y → x is much less than y |
| x ≤ y → x is less than or equal to y |
| x ∝ y → x is proportional to y |
| Basic Algebra: |
| (commutative) |
| (commutative) |
| (associative) |
| (associative) |
| (distributive) |
| Algebraic Equivalences: |
| A function denoted as f(x) is a relation f between a set of inputs x and a set of permissible outputs with the property that each input is related to exactly one output. |
| (even function) |
| (odd function) |
| (composition) |
| (non-commutative) |
| (inverse function f−1) |
| Common Functions: |
| → natural logarithm (base e) |
| → exponent ex |
| → floor or nearest lesser integer |
| → ceiling or nearest greater integer |
| → real part of complex number |
| → imaginary part of complex number |
| → absolute value or distance from origin |
| → Gamma function = (x-1)! |
| Exponents: |
| Exponents, also known as powers, are denoted with a superscript as in xn which is described as 'x to the power n'. This means 'x' multiplied by itself 'n' times. |
| (x squared) |
| (x cubed) |
| Roots: |
| Roots are denoted with the radical sign √ and are the result of taking fractional exponents. |
| (square root) |
| (cube root) |
| (nth root) |
| A logarithm is the inverse of exponent which is denoted logn x where n is called the base. |
| (natural log) |
| (common log) |
| Imaginary numbers are denoted xi where i is the imaginary constant and i2=−1. Complex numbers are denoted a+bi or a−bi; where i is the imaginary constant. This is usually called rectagular form. Complex and imaginary numbers can result from operations that have no real-number solutions such as square roots or logarithms of negative numbers. Complex numbers are typically represented as points on a plane where the x-axis is the real part and the y-axis is imaginary. |
| (conjugate) |
| (absolute value) |
| Alternatively, complex numbers can be represented in polar form denoted r∠θ where r (modulus or amplitude) is the magnitude and θ (argument or phase) is the angle with respect to the real axis within the complex plane. If z = a+bi then the polar form r∠θ is: |
| Trigonometric functions are defined as the ratios between two sides of a right triangle, denoted a and b called the legs, and c called the hypotenuse at a specific primary angle θ. All trig functions can be referenced to a unit circle with radius 1 on an X/Y axis, where the origin is defined as the center of the circle, the hypotenuse as the radius, the base as the x-axis, and the primary angle as the angle between the x-axis and the hypotenuse. | |||||||||||||||||||||||||||
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| Basic Identities: | |||||||||||||||||||||||||||
| (sine;cosine) | |||||||||||||||||||||||||||
| (tangent;cotangent) | |||||||||||||||||||||||||||
| (secant;cosecant) | |||||||||||||||||||||||||||
| Pythagorean Identities: | |||||||||||||||||||||||||||
| (Pythagorean Theorem) | |||||||||||||||||||||||||||
| Reciprocal Identities: | |||||||||||||||||||||||||||
| Quotient Identities: | |||||||||||||||||||||||||||
| Inverse Functions: | |||||||||||||||||||||||||||
| Inverse trig functions can be denoted using any of the following forms with the example of (sin θ): | |||||||||||||||||||||||||||
| (asin θ) or (sin-1 θ) or (arcsin θ) | |||||||||||||||||||||||||||
| Where: | |||||||||||||||||||||||||||
| ; −π/2 ≤ θ ≤ π/2 radians | |||||||||||||||||||||||||||
| ; 0 ≤ θ ≤ π radians | |||||||||||||||||||||||||||
| ; −π/2 ≤ θ ≤ π/2 radians | |||||||||||||||||||||||||||
| ; 0 ≤ θ ≤ π radians | |||||||||||||||||||||||||||
| ; 0 ≤ θ ≤ π radians | |||||||||||||||||||||||||||
| ; −π/2 ≤ θ ≤ π/2 radians | |||||||||||||||||||||||||||
| Exponential Forms(i2=-1): | |||||||||||||||||||||||||||
| (Euler's formula) | |||||||||||||||||||||||||||
| (Euler's identity) | |||||||||||||||||||||||||||
| Law of Cosines | |||||||||||||||||||||||||||
| where for any triangle: | |||||||||||||||||||||||||||
| a and b = adjacent sides | |||||||||||||||||||||||||||
| θ = angle between a and b | |||||||||||||||||||||||||||
| c = opposite side of angle θ | |||||||||||||||||||||||||||
| Law of Sines | |||||||||||||||||||||||||||
| where for any triangle: | |||||||||||||||||||||||||||
| A, B, and C are opposite angles of sides a, b, and c | |||||||||||||||||||||||||||
| Angles and Units: | |||||||||||||||||||||||||||
| 360 degrees = 2π radians = 400 grads | |||||||||||||||||||||||||||
|
| Hyperbolic functions are defined as analogs of trignometric functions except they are applied to hyperbolas instead of unit circles. As a result, they exhibit similar identities. Hyperbolic functions use the familiar trigonometric names with an 'h' suffix such as sinh, cosh, tanh. |
| Exponential Definitions: |
| Inverse Hyperbolic Functions: |
| Where: |
| ; −∞ < x < ∞ |
| ; 1 ≤ x < ∞ |
| ; 1 ≤ x < ∞ |
| ; 0 ≤ x ≤ 1 |
| ; −∞ < x < ∞ |
| ; −1 ≤ x ≤ 1 |
| Definitions: |
| Identities: |
| Relation to Trig Functions: |
| Summation: |
| Sums of discrete series are denoted using a capital Sigma (Σ), the lower expression defining the index variable and lower bound, the upper expression defining upper bound of integer values for the index. |
| (definition) |
| Products: |
| Products of discrete series are denoted using a capital Pi (Π), the lower expression defining the index variable and lower bound, the upper expression defining upper bound of integer values for the index. |
| (definition) |
| (factorial) |
| Taylor Series: |
| The Taylor Series is a method that allows any differentiable function to be expressed as a series. In many cases the Taylor series can have an infinite number of terms and the function can only be approximated via calculation. Expressed as a sum, the Taylor series is denoted: |
| where n! denotes the factorial of n and ƒ(n)(a) denotes the nth derivative of function ƒ evaluated at point a. |
| Maclaurin Series: |
| The Maclaurin Series is a special case of the Taylor Series that is evaluated at zero where the function has a solution at zero. This case significantly simplifies the construction of the polynomial series. A Maclaurin series can be constructed for many functions, which an infinite summation in the form: |
| where n! denotes the factorial of n and ƒ{n}(0) denotes the nth derivative of function ƒ evaluated at 0. |
| Common Maclaurin Series: |
| for |x| < 1 (geometric series) |
| Standard Deviation: |
| In statistics or probability theory, standard deviation represented by lower case sigma (σ), shows how much variation exists in a data set from the average or mean value. A small standard deviation indicates that the data points tend to be very close to the mean; a large standard deviation indicates that the data points are spread out over a wide range. |
σ = std. deviation of population S = std. deviation of sample xi = sample variable μ = arithmetic mean of population X = arithmetic mean of sample SE = Standard Error |
| Normal Distribution: |
| Log-Normal Distribution: |
σ = std. deviation x = random variable μ = population mean CDF = Cumulative Distribution Function PDF = Probability Distribution Function |
| Limits: |
| A limit is the output value that a function or sequence approaches, or converges to, as the input approaches some specific value. Limits are most useful when the input value approaches infinity, or approaches a divide-by-zero, and the limit is a finite quantity. |
| This example shows a limit when approaching a divide-by-zero condition: |
| This example shows a limit when the input value approaches infinity: |
| Limit of sum/difference: |
| Limit of product: |
| Limit of quotient: |
| Multiplication by constant: |
| Derivatives: |
| A derivative can be described as the slope of a point on a continuous function. Derivatives can represent instantaneous rates of change in any quantity that is described by a continuous function. For any function f(x)=y derivative is defined using the limit of the ratio of the change in y over the change in x (Δy/Δx) as Δx approaches zero. |
| The derivative of the function f(x) at the point x0 is denoted: |
| Second derivative: |
| Derivative of Sum/Difference: |
| Derivative of Product: |
| Derivative of Quotient: |
| Chain Rule: |
| Integrals: |
| An integral can be described as the area under a continuous curve on a graph. Integrals can represent the sum of all values over any quantity that is described by a continuous function. The integral is the inverse function of the derivative (antiderivative) as described by the Fundamental Theorem of Calculus. |
| Where: F'(x) = f(x) |
| - for a continuous function f(x), the indefinite integral is denoted: |
| - for a continuous function f(x) over interval [a,b], the definite integral is denoted: |
| Fundamental Theorem of Calculus: |
| Conventions: |
| If a ≤ b ≤ c : |
| Integral of Sum/Difference: |
| Multiplication by a constant: |
| Basics: |
| Trigonometric: |
| Inverse Trigonometric: |
| Hyperbolic: |
| Inverse Hyperbolic: |
| (Constant of integration ...+C assumed) |
| Basics: |
| Rational Functions: |
| Exponential: |
| Logarithmic: |
| Trigonometric: |
| Inverse Trigonometric: |
| Hyperbolic: |
| Inverse Hyperbolic: |