Every fraction has three signs (signs of fractions)

There are three signs associated with every fraction, one with the numerator, one with the denominator, and one with the fraction in general. But this can be hard to remember, because not all of the signs are always visible.

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Polar coordinates vs. rectangular coordinates

Any point in the coordinate plane can be expressed in both rectangular coordinates and polar coordinates. Instead of moving out from the origin using horizontal and vertical lines, like we would with rectangular coordinates, in polar coordinates we instead pick the angle, which is the direction, and then move out from the origin a certain distance.

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Vertical angles as congruent angles

Vertical angles are angles in opposite corners of intersecting lines. So vertical angles always share the same vertex, or corner point of the angle. They’re a special angle pair because their measures are always equal to one another, which means that vertical angles are congruent angles.

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How to get the domain and range from the graph of a function

The domain is all x-values or inputs of a function and the range is all y-values or outputs of a function. When looking at a graph, the domain is all the values of the graph from left to right. The range is all the values of the graph from down to up.

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Theorem of Pappus to find volume using the centroid

The Theorem of Pappus tells us that the volume of a three-dimensional solid object that’s created by rotating a two-dimensional shape around an axis is given by V=Ad. V is the volume of the three-dimensional object, A is the area of the two-dimensional figure being revolved, and d is the distance traveled by the centroid of the two-dimensional figure.

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Calculating absolute values

The absolute value operation turns any value inside it into its distance from the origin, essentially turning both positive and negative numbers into only positive numbers. Always calculate the value inside the absolute value first, then apply the absolute value last.

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Number sets in the real number system

The vast majority of the numbers you’ll use in most math classes are called real numbers, and the whole universe of real numbers is what makes up the Real Number System. Let’s start with a diagram.

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The alternating series estimation theorem to estimate the value of the series and state the error

The alternating series estimation theorem gives us a way to approximate the sum of an alternating series with a remainder or error that we can calculate. To use the theorem, the alternating series must follow two rules.

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Probability density functions and probability of X in an interval

Probability density refers to the probability that a continuous random variable X will exist within a set of conditions. It follows that using the probability density equations will tell us the likelihood of an X existing in the interval [a,b].

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Solving systems of three linear equations

Remember that a solution to a system of equations is the set of numbers that makes all of the equations true. If a three variable system has a solution, it’ll have a solution for each of the three variables.

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Intersecting tangents and secants of circles, intersecting inside the circle and outside the circle

There’s a special relationship between two secants that intersect outside of a circle. The length outside the circle, multiplied by the length of the whole secant is equal to the outside length of the other secant multiplied by the whole length of the other secant.

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Finding surface area of one function, bounded by another function, using a double integrals

You can use a double integral to find the area of a surface, bounded by another surface. The most difficult part of this will be finding the bounds of each of the integrals in the double integral.

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Solving limits with factoring

If you tried to solve the limit with substitution and it didn’t work, factoring should be the next thing you try. The goal will be to factor the function, and then cancel any removable discontinuities, in order to simplify the function, so that it can be evaluated.

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Solving initial value problems using laplace transforms

To use a Laplace transform to solve a second-order nonhomogeneous differential equations initial value problem, we’ll need to use a table of Laplace transforms or the definition of the Laplace transform to put the differential equation in terms of Y(s). Once we solve the resulting equation for Y(s), we’ll want to simplify it until we recognize that the terms in our equation match formulas in a table of Laplace transforms.

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How changes to the data change the mean, median, mode, range, and IQR

In this lesson, we want to see what happens to our measures of central tendency and spread when we make changes to our data set. Specifically the changes made either by changing all the values in the set at once, or by adding a single data point to, or removing a single data point from, the data set.

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Solving triple integrals using the midpoint rule

Similarly to the way we used midpoints to approximate single integrals by taking the midpoint at the top of each approximating rectangle, and to the way we used midpoints to approximate double integrals by taking the midpoint at the top of each approximating prism, we can use midpoints to approximate a triple integral by taking the midpoint of each sub-cube.

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How to calculate the differential of any multivariable function

Before we can use the formula for the differential, we need to find the partial derivatives of the function with respect to each variable. Then the differential for a multivariable function is given by three separate formulas.

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