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Calculus Help and Problems

This section contains in depth discussions and explanations on key topics that appear throughout Calculus 1 and 2 up through Vector Calculus. The topics are arranged in a natural progression catering typically to late highschool and early college students, covering the foundations of calculus, limits, derivatives, integrals, and vectors.

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Introduction to Calculus

Main Lesson: Introduction to Calculus

An introduction to Calculus and its applications.

Limits

Main Lesson: Limits

An overview of limits as it applies to differential and integral calculus.

Differential Calculus

Main Lesson: Differentiation - Taking the Derivative

An overview of Differential Calculus.

Integral Calculus

Main Lesson: Integration - Taking the Integral

An overview of Integral Calculus.

Multivariable / Vector Calculus

Main Lesson: Multivariable / Vector Calculus

Introduction to Calculus

Calculus is the study of change and motion, in the same way that geometry is the study of shape and algebra is the study of rules of operations and relations. It is the culmination of algebra, geometry, and trigonometry, which makes it the next step in a logical progression of mathematics.

Calculus defines and deals with limits, derivatives, and integrals of functions. The key ingredient in calculus is the notion of infinity. The essential link to completing calculus and satisfying concerns about infinite behavior is the concept of the limit, which lays the foundation for both derivatives and integrals.

Calculus is often divided into two sections: Differential Calculus (dealing with derivatives, e.g. rates of change and tangents), and Integral Calculus (dealing with integrals, e.g. areas and volumes). Differential Calculus and Integral Calculus are closely related as we will see in subsequent pages. It is important to have a conceptual idea of what calculus is and why it is important in order to understand how calculus works.

History of Calculus

Main Lesson: Brief History of Calculus

A brief history of the invention of calculus and its development.

Difference Between Calculus and Other Mathematics

Main Lesson: Difference Between Calculus and Other Mathematics

A comparison of calculus against other mathematical disciplines.

Calculus Applications in Algebra and Geometry

Main Lesson: Calculus Applications in Algebra and Geometry

A Brief History of Calculus

Calculus was created by Isaac Newton, a British scientist, as well as Gottfried Leibniz, a self-taught German mathematician, in the 17th century. It has been long disputed who should take credit for inventing calculus first, but both independently made discoveries that led to what we know now as calculus. Newton discovered the inverse relationship between the derivative (slope of a curve) and the integral (the area beneath it), which deemed him as the creator of calculus. Thereafter, calculus was actively used to solve the major scientific dilemmas of the time, such as:

• calculating the slope of the tangent line to a curve at any point along its length
• determining the velocity and acceleration of an object given a function describing its position, and designing such a position function given the object's velocity or acceleration
• calculating arc lengths and the volume and surface area of solids
• calculating the relative and absolute extrema of objects, especially projectiles

For Newton, the applications for calculus were geometrical and related to the physical world - such as describing the orbit of the planets around the sun. For Leibniz, calculus was more about analysis of change in graphs. Leibniz's work was just as important as Newton's, and many of his notations are used today, such as the notations for taking the derivative and the integral.

Difference Between Calculus and Other Math Subjects

On the left, a man is pushing a crate up a straight incline. On the right, a man is pushing the same crate up a curving incline. The problem in both cases is to determine the amount of energy required to push the crate to the top. For the problem on the left, you can use algebra and trigonometry to solve the problem. For the problem on the right, you need calculus. Why do you need calculus with the problem on the right and not the left?

This is because with the straight incline, the man pushes with an unchanging force and the crate goes up the incline at an unchanging speed. With the curved incline on the right, things are constantly changing. Since the steepness of the incline is constantly changing, the amount of energy expended is also changing. This is why calculus is described as "the mathematics of change". Calculus takes regular rules of math and applies them to evolving problems.

With the curving incline problem, the algebra and trigonometry that you use is the same, the difference is that you have to break up the curving incline problem into smaller chunks and do each chunk separately. When zooming in on a small portion of the curving incline, it looks as if it is a straight line:

Then, because it is straight, you can solve the small chunk just like the straight incline problem. When all of the small chunks are solved, you can just add them up.

This is basically the way calculus works - it takes problems that cannot be done with regular math because things are constantly changing, zooms in on the changing curve until it becomes straight, and then it lets regular math finish off the problem. What makes calculus such a brilliant achievement is that it actually zooms in infinitely. In fact, everything you do in calculus involves infinity in one way or another, because if something is constantly changing, it is changing infinitely from each infinitesimal moment to the next. All of calculus relies on the fundamental principle that you can always use approximations of increasing accuracy to find the exact answer. Just like you can approximate a curve by a series of straight lines, you can also approximate a spherical solid by a series of cubes that fit inside the sphere.

Algebra and Geometry with Calculus

One of the earliest algebra topics learned is how to find the slope of a line--a numerical value that describes just how slanted that line is. Calculus gives us a much more generalized method of finding slopes. With it, we can find not only how steeply a line slopes, but indeed, how steeply any curve slopes at any given point.

Without calculus, it is difficult to find areas of shapes other than those whose formulas you learned in geometry. You may be able to find the area of commons shapes such as a triangle, square, rectangle, circle, and even a trapezoid; but how could you find the area of the shape like the one shown below?

With calculus, you can calculate complicated x-intercepts. Without a graphing calculator, it is pretty difficult to calculate an irrational root. However, a simple process called Newton's Method (named Isaac Newton) allows you to calculate an irrational root to whatever accuracy you want.

Calculus makes it much easier to visualize graphs. You may already have a good grasp of linear functions and how to visualize their graphs easily, but what about the graph of something like y= x^3 + 2x^2 - x + 1? Elementary calculus tells you exactly where that graph will be increasing, decreasing, and twisting. You can even find the highest and lowest points on the graph without plotting a single point.

One of the most useful applications of calculus is the optimization of functions. In a small number of steps, you can answer questions such as:

If I have 500 feet of fence, what is the largest rectangular yard I can make? or Given a rectangular sheet of paper which measures 8.5 inches by 11 inches, what are the dimensions of the box I can make containing the greatest volume?

The traditional way to create an open box from a rectangular surface is to cut congruent squares from the corners of the rectangle and then to fold the resulting sides up as shown:

Calculus develops concepts in other mathematics that lets us discover more about them and enables us to achieve greater feats than the mathematics that it is built on. It is vital to understanding and making sense of the world we live in.

Applications of Calculus

With calculus, we have the ability to find the effects of changing conditions on a system. By studying these, you can learn how to control a system to make it do what you want it to do. Because of the ability to model and control systems, calculus gives us extraordinary power over the material world.

Calculus is the language of engineers, scientists, and economists. The work of these professionals has a huge impact on our daily life - from your microwaves, cell phones, TV, and car to medicine, economy, and national defense.

Credit card companies use calculus to set the minimum payments due on credit card statements at the exact time the statement is processed by considering multiple variables such as changing interest rates and a fluctuating available balance.

Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed. This research can help increase the rate of growth of necessary bacteria, or decrease the rate of growth for harmful and potentially threatening bacteria.

An electrical engineer uses integration to determine the exact length of power cable needed to connect two substations that are miles apart. Because the cable is hung from poles, it is constantly curving. Calculus allows a precise figure to be determined.

An architect will use integration to determine the amount of materials necessary to construct a curved dome over a new sports arena, as well as calculate the weight of that dome and determine the type of support structure required.

Space flight engineers frequently use calculus when planning lengthy missions. To launch an exploratory probe, they must consider the different orbiting velocities of the Earth and the planet the probe is targeted for, as well as other gravitational influences like the sun and the moon. Calculus allows each of those variables to be accurately taken into account.

Statisticians will use calculus to evaluate survey data to help develop business plans for different companies. Because a survey involves many different questions with a range of possible answers, calculus allows a more accurate prediction for appropriate action.

A physicist uses calculus to find the center of mass of a sports utility vehicle to design appropriate safety features that must adhere to federal specifications on different road surfaces and at different speeds.

An operations research analyst will use calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operating efficiency, increase production, and raise profits.

A graphics artist uses calculus to determine how different three-dimensional models will behave when subjected to rapidly changing conditions. This can create a realistic environment for movies or video games.

Obviously, a wide variety of careers regularly use calculus. Universities, the military, government agencies, airlines, entertainment studios, software companies, and construction companies are only a few employers who seek individuals with a solid knowledge of calculus. Even doctors and lawyers use calculus to help build the discipline necessary for solving complex problems, such as diagnosing patients or planning a prosecution case. Despite its mystique as a more complex branch of mathematics, calculus touches our lives each day, in ways too numerous to calculate.

Help with Limits in Calculus

All of calculus relies on the principle that we can always use approximations of increasing accuracy to find the exact answer, such as approximating a curve by a series of straight lines in differential calculus (the shorter the lines and as the distance between points approaches 0, the closer they are to resembling the curve) or approximating a spherical solid by a series of cubes in integral calculus (as the size of the cubes gets smaller and the number of cubes approaches infinity inside the sphere, the end result becomes closer to the actual area of the sphere).

With the help of modern technology, graphs of functions are often easy to produce. The main focus is between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and long term behavior of a function. In Calculus classes, limits are usually the first topic introduced.

In order to understand the workings of differential and integral calculus, we need to understand the concept of a limit. Limits are used in differentiation when finding the approximation for the slope of a line at a particular point, as well as integration when finding the area under a curve. In calculus, limits introduce the component of infinity. We can ask ourselves, what happens to the value of a function as the independent variable gets infinitely close to a particular value?

The graph illustrates finding the limit of the dependent variable f(x) as x approaches c. A way to find this is to plug in values that gets close to c from the left and values close to c from the right.

To further illustrate the concept of a limit, consider the sequence of numbers of x:

These values are getting closer and closer to 2 (i.e. they are approaching 2 as their limit). We can can say that no matter what value we consider, 2 is the smallest value that is greater than every output f(x) in the sequence. As we take the differences of these numbers, they will get smaller and smaller. In calculus, the difference between the terms of the sequence and their limit can be made infinitesimally small.

Sometimes, finding the limiting value of an expression means simply substituting a number.

(1) Find the limit as t approaches 10 of the expression

We write this using limit notation as

In this example, we simply substitute and write

There is no complication because M = 3t + 7 is a continuous function, but there are cases where we cannot simply substitute like this.

(2) Find the limit as x approaches 0 of

Continuity and Limits

Many theorems in calculus require that functions be continuous on intervals of real numbers. To successfully carry out differentiation and integration over an interval, it is important to make sure the function is continuous.

Definition

A function f is continuous at a point (c, f(c)) if all three conditions are satisfied:

1) An output of c exists: