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A detailed tutorial on revenue, cost, and product functions. Step by step tutorial including several examples of revenue, cost, and product functions for reference.

The revenue, cost, and product functions are parts of economics and business math. The cost function is how much something costs, and it can be expressed as C(q) = 100 + 2q. The revenue function is how much money you get from selling what you bought, and it can be expressed as R(q) = 2.5q. The profit function is how much money was actually made, and it is the revenue function minus the cost function.

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A detailed tutorial on writing proofs by exhaustion. Step by step tutorial including several examples of how to write proofs by exhaustion for reference.

A proof by exhaustion is one of the easier types of proofs to write. All this proof involves is testing cases – every case possible for what you are trying to prove. This can be made easier by using variables instead of numbers, or by testing for an even number and odd number, positive and negative number, etc. That way you do not have to test many numbers in order to prove. If even one of the cases does not work out, then whatever you are testing for has been disproven.

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A detailed tutorial on how to solve for Mach numbers. Step by step tutorial including several examples of how to solve for Mach numbers for reference.

A Mach number is the speed of an object moving through the air, or any fluid substance, divided by the speed of sound as it is in that substance. It is often used to represent an object such as an aircraft or a missile’s speed, when it is travelling at the speed of sound or multiples of the speed of sound. This can be portrayed mathematically in the equation M = vs / u, where M is the Mach number, vs is the speed of the source (the object relative to the medium), and u is the speed of sound in the medium.

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A detailed tutorial on how to find values of quadrantal angles. Step by step tutorial including several examples of finding values of quadrantal angles for reference.

Quadrantal angles have a terminal side coinciding with a coordinate axis. A trigonometric functional value of such an angle can be determined by the coordinates of the point where the terminal side intersects the unit circle. When on the unit circle, the Cartesian coordinate (x, y) cooresponds to (cos(&), sin(&)) on the unit circle.

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A detailed tutorial on identifying coterminal angles. Step by step tutorial including several examples of how to identify coterminal angles for reference.

Coterminal angles are opposite angles that when put together share a terminal side, or common side, and therefore create a circle. One of the angles is positive, and the other angle is negative – a negative angle is one that is formed from the opposite side and using the second scale on a protractor. The absolute value of the first angle plus the absolute value of the second angle must add up to 360 degrees in order for them to be coterminal angles.

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A detailed tutorial on the conjugate zeros theorem. Step by step tutorial including several examples of the conjugate zeros theorem for reference.

The conjugate zeros theorem states that if a + b * i is a zero of a polynomial with real coefficients, then so is a – b * i. The conjugate zeros theorem can be proved by taking any function in this form and setting it equal to zero. The conjugate zeros theorem makes many equations easier to solve, especially complex equations when you get to higher levels of math.

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A detailed tutorial on how to find the directrix of a parabola. Step by step tutorial including several examples of how to find the directrix of a parabola for reference.

A parabola is a curved shape that is formed by the graph of the function x squared. A parabola is technically known as the locus of points where the distance to the focus equals the distance to the directrix. The directrix is a given line on a parabola that does not go through the focus.

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A detailed tutorial on the inverse matrix and how to calculate the inverse matrix. Step by step tutorial including several examples of the inverse matrix for reference.

All square matrices have an inverse, except for the rare invertible matrices, called singular matrices. The inverse of a square matrix can be defined in mathematical terms as the matrix times the inverse of the matrix is equal to I, which represents the identity matrix. The inverse of a matrix may be found by using the inverse function. This makes the inverse easy to find, as you follow basic rules for finding the inverse of other types of equations.

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A detailed tutorial on singular matrices. Step by step tutorial including several examples of singular matrices and how to identify singular matrices  for reference.

A singular matrix is a square matrix that is not invertible. In order to not be invertible, the determinant must be zero. No other values will make a matrix singular. Single matrices are very rare – almost all square matrices are invertible. A quick way to find out if a matrix is invertible or singular is to attempt to invert the matrix.

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A detailed tutorial on infinite sets. Step by step tutorial including several examples of infinite sets and how to identify them for reference.

There are two types of sets, finite sets and infinite sets. The tutorial will focus on infinite sets. An infinite set is a set that has at least one endpoint of infinity, which can be implied either by having infinity in the set or by having a trailing end of the set, with no number at the end. Infinite sets can either be countable or uncountable – meaning they either have a pattern you can use to follow to infinity, or there is no pattern present.