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How to Vertically Transform a Function

To transform a function vertically, you add a number to or subtract a number from the entire function, or multiply it by a number. To do something to an entire function, say [more…]

How to Solve Limits by Conjugate Multiplication

To solve certain limit problems, you’ll need the conjugate multiplication technique. When substitution doesn’t work in the original function — usually because of a hole in the function — you can use conjugate [more…]

How to Solve Limits at Infinity by Using Horizontal Asymptotes

Horizontal asymptotes and limits at infinity always go hand in hand. You can’t have one without the other. If you’ve got a rational function like [more…]

The Basic Differentiation Rules

Some differentiation rules are a snap to remember and use. These include the constant rule, power rule, constant multiple rule, sum rule, and difference rule. [more…]

How to Work with Lines in Calculus

A line is the simplest function that you can graph on the coordinate plane. (Lines are important in calculus because when you zoom in far enough on a curve, it looks and behaves like a line.) This figure [more…]

How to Recognize Inverse Functions

You can tell that two functions are inverse functions when each one undoes what the other does. When you graph inverse functions, each is a mirror image of the other. Here are some examples of inverse [more…]

How to Use SohCahToa to Find the Trig Functions of a Right Triangle

The study of trigonometry begins with the right triangle. The three main trig functions (sine, cosine, and tangent) and their reciprocals (cosecant, secant, and cotangent) all tell you something about [more…]

How to Graph Sine, Cosine, and Tangent

So, you need to graph a sine, cosine, or tangent function. Sine, cosine, and tangent — and their reciprocals, cosecant, secant, and cotangent — are periodic [more…]

How to Use Limits to Determine Continuity

Here you’ll learn about continuity for a bit, then go on to the connection between continuity and limits, and finally move on to the formal definition of continuity. [more…]

Solving Easy Limits

There are two types of easy limit problems: the ones you should just memorize and the ones where you can plug in the x-number and get the answer in one step. [more…]

How to Solve Limits with a Calculator

You can solve most limit problems by using your calculator. There are two basic methods. For example, say you want to evaluate the following limit: [more…]

How to Solve Limits by Factoring

You can use the algebraic technique of factoring to solve “real” limit problems. All algebraic methods involve the same basic idea. When substitution doesn’t work in the original function — usually because [more…]

How to Solve Limits with Basic Algebra

When substitution doesn’t work in the original limit function — usually because of a hole in the function — you can often use some algebra to manipulate the function until substitution does work [more…]

How to Solve Limits at Infinity with a Calculator

Solving for limits at infinity is easy to do when you use a calculator. For example, enter the below function in your calculator's graphing mode: [more…]

How to Solve Limits at Infinity by Using Algebra

Yes, you can solve a limit at infinity using a calculator, but all things being equal, it’s better to solve the problem algebraically, because then you have a mathematically airtight answer. For example [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 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…]

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…]

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