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### Expressing Functions as Power Series Using the Maclaurin Series

The *Maclaurin series* is a template that allows you to express many other functions as power series. It is the source of formulas for expressing both sin [more…]

### Expressing Functions as Power Series Using the Taylor Series

The Taylor series provides a template for representing a wide variety of functions as power series. It is relatively simple to work with, and you can tailor it to obtain a good approximation of many functions [more…]

### Determining Whether a Taylor Series Is Convergent or Divergent

Because the Taylor series is a form of power series, every Taylor series also has an interval of convergence. When this interval is the entire set of real numbers, you can use the series to find the value [more…]

### Calculating Error Bounds for Taylor Polynomials

A Taylor polynomial approximates the value of a function, and in many cases, it’s helpful to measure the accuracy of an approximation. This information is provided by the [more…]

### Calculating Magnitude with Vectors

Vectors are commonly used to model forces such as wind, sea current, gravity, and electromagnetism. Calculating the magnitude of vectors is essential for all sorts of problems where forces collide. [more…]

### Using Scalar Multiplication with Vectors

Multiplying a vector by a scalar is called *scalar multiplication.* To perform scalar multiplication, you need to multiply the scalar by each component of the vector. [more…]

### Finding the Unit Vector of a Vector

Every nonzero vector has a corresponding *unit vector,*which has the same direction as that vector but a magnitude of 1. To find the unit vector **u** of the vector [more…]

### Adding and Subtracting Vectors

You can add and subtract vectors on a graph by beginning one vector at the endpoint of another vector. You add and subtract vectors component by component, as follows: [more…]

### How to Plot Cylindrical Coordinates

*Cylindrical coordinates* are simply polar coordinates with the addition of a vertical *z*-axis extending from the origin. While a polar coordinate pair is of the form [more…]

### How to Plot Spherical Coordinates

*Spherical coordinates* are used — with slight variation — to measure latitude, longitude, and altitude on the most important sphere of them all, the planet Earth. Every point in space is assigned a set [more…]

### How to Use a Partial Derivative to Measure a Slope in Three Dimensions

You can use a partial derivative to measure a rate of change in a coordinate direction in three dimensions. To do this, you visualize a function of two variables [more…]

### Determine Signed Areas in a Problem

The solution to a definite integral gives you the *signed*area of a region. In some cases, signed area is what you want, but in some problems you’re looking for [more…]

### Determine Unsigned Area between Curves

You can use the concept of unsigned area to measure the area between curves. For example, you can use this technique to find the unsigned shaded area in the following figure. [more…]

### Integrating Powers of Cotangents and Cosecants

You can integrate powers of cotangents and cosecants similar to the way you do tangents and secant. For example, here’s how to integrate cot^{8} *x* csc^{6} *x:* [more…]

### Setting Up Partial Fractions When You Have Repeated Quadratic Factors

Your first step in any problem that involves partial fractions is to recognize which case you’re dealing with so that you can solve the problem. One case where you can use partial fractions is with repeated [more…]

### Pre-Calculus Unit Circle

In pre-calculus, the unit circle is sort of like unit streets, it’s the very small circle on a graph that encompasses the 0,0 coordinates. It has a radius of 1, hence the unit. The figure here shows all [more…]

### Right Triangles and Trig Functions for Pre-Calculus

If you’re studying pre-calculus, you’re going to encounter triangles, and certainly the Pythagorean theorem. The theorem and how it applies to special right triangles are set out here: [more…]

### How to Format Interval Notation in Pre-Calculus

In pre-calculus you deal with inequalities and you use interval notation to express the solution set to an inequality. The following formulas show how to format solution sets in interval notation. [more…]

### Absolute Value Formulas for Pre-Calculus

Even though you’re involved with pre-calculus, you remember your old love, algebra, and that fact that absolute values then usually had two possible solutions. Now that you’re with pre-calculus, you realize [more…]

### Trig Identities for Pre-Calculus

Of course you use trigonometry, commonly called trig, in pre-calculus. And you use trig identities as constants throughout an equation to help you solve problems. The always-true, never-changing trig identities [more…]

### Pre-Calculus For Dummies Cheat Sheet

Pre-Calculus bridges Algebra II and Calculus. Pre-calculus involves graphing, dealing with angles and geometric shapes such as circles and triangles, and finding absolute values. You discover new ways [more…]

### How to Graph a Rational Function with Numerator and Denominator of Equal Degrees

After you calculate all the asymptotes and the *x-* and *y-*intercepts for a rational function, you have all the information you need to start graphing the function. Rational functions with equal degrees in [more…]

### Understanding the Properties of Numbers

Remembering the properties of numbers is important because you use them consistently in pre-calculus. The properties aren’t often used by name in pre-calculus, but you’re supposed to know when you need [more…]

### How to Graph Linear Inequalities

You can use the slope-intercept form to graph inequalities. The *slope-intercept form* is expressed as *y = mx + b*, where the variable *m* stands for the slope of the line, and [more…]

### How to Use a Graphing Calculator

It’s a good idea to purchase a graphing calculator for pre-calculus work. Since the invention of the graphing calculator, math classes have begun to change their scope. A graphing calculator does so many [more…]