How to find Mean, variance, and standard deviation

It’s important to know whether we’re talking about a population or a sample, because in this section we’ll be talking about variance and standard deviation, and we’ll use different formulas for variance and standard deviation depending on whether we’re using data from a population or data from a sample.

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Position functions and velocity and acceleration

You’re usually given a position equation x or s(t), which tells you the object’s distance from some reference point. This equation also accounts for direction, so the distance could be negative, depending on which direction your object moved away from the reference point.

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How to find the error or remainder of a series

If you need to find the sum of a series, but you don’t have a formula that you can use to do it, you can instead add the first several terms, and then use the integral test to estimate the very small remainder made up by the rest of the infinite series. The sum of the series is usually the sum of the first several terms, plus a very smaller error that you can estimate.

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Partial derivatives in two variable functions

Finding derivatives of a multivariable function means we’re going to take the derivative with respect to one variable at a time. For example, we’ll take the derivative with respect to x while we treat y as a constant, then we’ll take another derivative of the original function, this one with respect to y while we treat x as a constant.

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Symmetric equations for the line of intersection of two planes

If two planes intersect each other, the curve of intersection will always be a line. To find the symmetric equations that represent that intersection line, you’ll need the cross product of the normal vectors of the two planes, as well as a point on the line of intersection.

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Finding inverse trig derivatives

To avoid confusion between negative exponents and inverse functions, sometimes it’s safer to write arcsin instead of sin^(-1) when you’re talking about the inverse sine function. The same thinking applies to the other five inverse trig functions.

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How to add and subtract rational expressions

We’ll look at how to add and subtract fractions that have variables in them as well as numbers. We’ll start by finding a common denominator from the factors of each of the individual denominators.

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How to find the orthogonal trajectories

The orthogonal trajectories to a family of curves are the curves that intersect each member of the family at a perfectly perpendicular angle. So given a family of curves, you can change the value of the constant in the equation that models the family, to create a family of many curves, and then sketch the family in the plane.

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