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Algebra
How to Add and Subtract Fractions in Algebra (Video)
In algebra, adding and subtracting fractions is easy when you find the common denominator. This video shows you how to convert fractions for the common denominator. After you determine the common denominator, you can add and subtract fractions, including story problems, with ease. [more…]
Calculus
How to Find the Volume of a Solid with a Circular Cross-Section (Video)
Calculus allows you to calculate the volume of conical objects by dividing the object into an infinite number of circular cross-sections - geometrical shapes resembling pancakes or washers - and adding up the volume of all those cross-sections through integration. This video tutorial shows you how. [more…]
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Geometry
Mastering the Formal Geometry Proof
Suppose you need to solve a crime mystery. You survey the crime scene, gather the facts, and write them down in your memo pad. To solve the crime, you take the known facts and, step by step, show who committed [more…]
Geometry
Sizing Up the Area of a Polygon
Not only can polygons be classified by the number of sides they have and by their angles, but they can also be grouped according to some of their qualities. Polygons can have three personality characteristics [more…]
Geometry
Simple and Easy Geometry Tips and Tools
The first rule of life? Life (as well as geometry) can be difficult. But why make it more difficult than it has to be? Do you need help with geometry? Here are 11 tried-and-true tips to make your forays [more…]
Geometry
Measuring and Making Angles
On a map, you trace your route and come to a fork in the road. Two diverging roads split from a common point and form an angle. The point at which the roads diverge is the [more…]
Geometry
Classifying Three Types of Triangles
Triangles are classified according to the length of their sides or the measure of their angles. These classifications come in threes, just like the sides and angles themselves. That is, a triangle has [more…]
How to Find the Derivative of a Line
The derivative is just a fancy calculus term for a simple idea that you probably know from algebra — slope. Slope is the fancy algebra term for steepness. And [more…]
How to Find the Derivative of a Curve
Calculus is the mathematics of change — so you need to know how to find the derivative of a parabola,which is a curve with a constantly changing slope. [more…]
How to Differentiate the Trigonometric Functions
You should memorize the derivatives of the six trig functions. Make sure you memorize the first two in the following list — they’re a snap. If you’re good at rote memorization, memorize the last four as [more…]
How to Differentiate Exponential and Logarithmic Functions
Differentiating exponential and logarithmic functions involves special rules. No worries — once you memorize a couple of rules, differentiating these functions is a piece of cake. [more…]
How to Find Derivatives Using the Product and Quotient Rules
There are special rules for finding the derivative of the product of two functions or the quotient of two functions; these are the product rule and the quotient rule, respectively. [more…]
How to Differentiate Implicitly
In most differentiation problems, y is written explicitly as a function of x, which just means that the equation is solved for y; in other words, y is by itself on one side of the equation. Sometimes, [more…]
How to Use Logarithmic Differentiation
For differentiating certain functions, logarithmic differentiation is a great shortcut. It spares you the headache of using the product rule or of multiplying the whole thing out and then differentiating [more…]
How to Differentiate Inverse Functions
There’s a difficult-looking formula involving the derivatives of inverse functions, but before you get to that, look at the following figure, which nicely sums up the whole idea. [more…]
How to Find High-Order Derivatives
Finding a second, third, fourth, or higher derivative is incredibly simple. The second derivative of a function is just the derivative of its first derivative. The third derivative is the derivative of [more…]
How to Find a Function’s Absolute Extrema over Its Entire Domain
A function’s absolute max and absolute min over its entire domain are the single highest and single lowest values of the function anywhere it’s defined. A function can have an absolute max or min or both [more…]
The Mean Value Theorem
You don’t need the mean value theorem for much, but it’s a famous theorem — one of the two or three most important in all of calculus — so you really should learn it. Fortunately, it’s very simple. [more…]
How to Use Differentiation to Calculate the Maximum Volume of a Box
One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function. In the following example, you calculate the maximum volume of a box that has no top and [more…]
How to Use Differentiation to Calculate the Maximum Area of a Corral
Finding the maximum or minimum value of a real-world function is one of the most practical uses of differentiation. For example, you might need to find the maximum area of a corral, given a certain length [more…]
Related Rates: the Expanding Balloon Problem
Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]
Related Rates: the Trough of Swill Problem
Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]
Related Rates: Two Cars at a Crossroads
Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]
How to Find the Tangent Lines of a Parabola that Pass through a Certain Point
Ever want to determine the location of a line through a given point that’s tangent to a given curve? Of course you have! Here’s how you do it.
Determine the points of tangency of the lines through the point [more…]
How to Find a Normal Line to a Curve
A line normal to a curve at a given point is the line perpendicular to the line that’s tangent at that same point. Find the points of perpendicularity for all normal lines to the parabola [more…]
How to Determine Marginal Cost, Marginal Revenue, and Marginal Profit in Economics
Marginal cost, marginal revenue, and marginal profit all involve how much a function goes up (or down) as you go over 1 to the right — this is very similar to the way linear approximation works. [more…]
How to Approximate Area with Left Rectangles
You can approximate the area under a curve by summing up “left” rectangles. For example, say you want the area under the curve f (x) = x2 + 1 from 0 to 3. The shaded area of the graph on the left side [more…]
How to Approximate Area with Midpoint Rectangles
A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangle's top side. A midpoint sum is a much better estimate of area than either a left-rectangle [more…]
How to Approximate Area with the Trapezoid Rule
With the trapezoid rule, instead of approximating area by using rectangles (as you do with the left, right, and midpoint rectangle methods), you approximate area with — can you guess? — trapezoids. [more…]
How to Approximate Area with Simpson's Rule
With Simpson’s rule, you approximate the area under a curve with curvy-topped “trapezoids.” The tops of these shapes are sections of parabolas. You can call them “trapezoids” because they play the same [more…]
How to Do Integration by Parts
Integrating by parts is the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you [more…]
How to Solve (and Factor) a Quadratic Equation with the Quadratic Formula
A quadratic equation is any second-degree polynomial equation — that’s when the highest power of x,or whatever other variable is used, is 2. The solution or solutions of a quadratic equation, [more…]
