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 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…]
