Whether you’re starting your first semester of calculus, or cramming for your final exam, integralCALC will help you quickly build skills with simple, step-by-step lessons.
Krista King (founder of integralCALC) worked as a calculus tutor while she was a student at the University of Notre Dame. After graduation, she started integralCALC.com to serve as a resource for calculus students. Since the beginning of 2010, Krista’s calculus lessons have helped students around the world and she continues to publish lessons regularly.
Lost in the world of calculus? Start at the beginning with integralCALC’s online tutorials. The first in a series of eight, this lesson introduces the idea of functions - number ‘machines’ that describe a relationship between two or more variables (like ‘x’ and ‘y'). Learn how functions differ from equations, what it means to find a function’s domain and range, and how to use the vertical line test to check a function’s validity.
You've got a graph, but how do you know whether or not it represents a function? Use the vertical line test! This lesson from IntegralCALC's Calculus I Essentials series teaches you all about the vertical line test: if you draw a vertical line anywhere on the graph and it crosses more than two points, then the graph fails the test and is not a function. This handy mathematical trick even can apply to discontinuous graphs!
Calculus can be frustrating if you don’t have a thorough grasp of the concepts. Build your confidence and improve your test scores with lessons from integralCALC! In this lesson, Krista discusses limits: learn how to solve for them, as well as three helpful shortcuts to help you find infinite limits. Stick around for the whole lesson and watch Krista demonstrate what to do in the event of a point discontinuity or a jump discontinuity. Calculus doesn’t have to be difficult—demystify limits!
Take your math skills to the limit! The general limit exists only if the left-hand and right-hand limits exist, and the two are equal. This lesson from IntegralCALC's Calculus I Essentials series teaches you how to find the left-hand and right-hand limits (negative and positive limits), and then use those limits to prove that the general limit does not exist. Although this lesson uses a specific example, the technique shown here can be applied to finding or disproving the limit of any formula.
Know your limits! But what is a limit in mathematical terms? This lesson from IntegralCALC's Calculus I Essentials series introduces you to the precise definition (or epsilon delta definition) of a limit. The introductory video explains the definition in detail, and then walks you through the step-by-step process of using that definition to prove a limit is true.
Do you know how to answer one of the most fundamentally important questions in calculus, and find the rate of change of a function at a point? In this lesson on derivatives from integralCALC, Krista explains what the rate of change is (the slope of a line), and how to use lines tangent and secant to a point on a curved line to calculate the derivative. With Krista’s help, you’ll be able to calculate the average and instantaneous rate of change in no time at all!
Learn to divide out derivatives with help from this video on evaluating difference quotients. This lesson from IntegralCALC’s Calculus I Essentials series teaches you how to use the definition of a derivative to find the value of a derivative at a particular point. See how to plug in values and functions, simplify your formula, and then cancel out variables and numbers to make solving a confusing equation easy.
Learn how to find the equation of a line that is tangent to a particular function! In this example from IntegralCalc, Krista shows you how to begin plugging in known variables. Then, Krista demonstrates how to simplify the equation once you’ve got your formulas and variables figured out, and eliminate the variable in the denominator of the equation. Next, learn how simplify to find the slope, and the point on the line. With the slope and a point, finding the final answer couldn’t be simpler!
Implicit differentiation can be tricky, so improve your understanding of the idea and work through an example equation with Krista King from IntegralCalc. Krista shows how to take the derivative of both sides of the equation, starting with the left. Then, learn to use the product and chain rule rules. Learn when you must multiply by dy/dx. Once you’ve simplified the equation and separated terms, you’ll be left with your final answer. With Krista’s help, implicit differentiation is simple!
Optimization is all about finding the extremes of a function—the high and low points. In this lesson from integralCALC, Krista demonstrates how to solve optimization problems in five easy steps, starting with taking the derivative, and ending by plugging the critical points back into the function. To finish, Krista uses optimization in a real world application, demonstrating how to optimize the surface area of an open topped box with known and unknown dimensions.
One of the most difficult and most common applications of derivatives is the related rates problem. But related rates don’t have to be hard! In this lesson from integralCALC, Krista gives you the tools to conquer these problems. Implicit differentiation is the key to learning how to deal with an equation when there is no clean separation of the unknown variables—once you’ve got that down, you’ll be able to solve any related rates problem in just four easy steps.
Not sure if your function is continuous? Learn to identify and solve for removable discontinuities with integralCALC as she walks through step-by-step instructions for finding values for which a function is undefined, and determining whether those values represent removable discontinuities or vertical asymptotes. She’ll also teach you how to solve for the y-value that will ‘plug the hole’ in your graph. Sometimes all you need is to see someone walk through an example start to finish!
In the previous lesson, Krista demonstrated how to calculate the derivative of a function by taking the definition of the derivative—in this lesson, learn tricks and tips to make calculating the derivative even faster and easier! Krista shows you how to use the power, product, and quotient rules to find the derivative of polynomial functions, the product of two functions, and fractions or quotients. Though these are the main rules you’ll use, Krista also shows you how to use the chain rule!
This example problem has it all! Learn to take the derivative of a complex natural log using the chain rule and tips from your algebra days. Follow along with integralCALC as she walks through a multi-level problem that illustrates how to apply the chain rule and natural log formula, and reminds you of how to simplify terms in complex fractions. Sometimes all you need is to see someone walk through an example start to finish!
Stumped by integral notation? Wonder how to denote the constant that would disappear when taking the derivative of a function? Just plain confused about calculus? In this lesson on integrals from Krista King of IntegralCALC, build off of your knowledge of derivatives to understand how to calculate an “antiderivative” to find the exact area under a function.
If you aren’t already a pro at solving integrals, things are about to change. Learn from Krista King how to solve an integral using three different techniques; U-substitution (Krista says this is usually the easiest and fastest way), integration by parts, or partial fractions (try this method first if you have any rational functions). Calculus just got easier!
Having a hard time applying the average value formula? Follow along with integralCALC as she walks through an example problem with step-by-step instructions on how to calculate the average value of a function. This lesson will remind you how to set up and evaluate the average value formula, as well as how to use u-substitution to more easily solve complicated integrals.