# Calc101.com FAQs

### Q: What does calc101.com do for me?

A: You get step-by-step solutions for derivatives and indefinite integrals in a completely automatic way, with each step explained.

You also get step-by-step determinants and matrix inverses.

And you can get the graph of a function automatically!

### Q: How does that help me?

A: You can get as many fully worked out solutions as you need.

After a while you'll master the basic computations and be free to understand the concepts. You'll have the confidence to tackle applications like related rates or volumes of revolution without fear.

It's like anything else: get a good foundation and build on it.

Yes, tutoring is now available. Email for information.

### Q: What do I have to do to get solutions?

A: Derivatives are free! Enter your function in the derivative field and press Do it!

You get the second derivative as well as the first. You can take the second derivative with respect to a different variable.

Polynomial long multiplication and long division are also free.

Integrals are cheap. You need to get a password to do an integral. You can buy a \$25 password for 900 solutions.

 Go to the integrals page. Enter your password. Enter your function. Enter a new variable if it is different from x. Press the "Do it" button.
You can come back any time to do more.

The same ideas apply to the curve sketching and step-by-step determinants and matrix inverses.

### Q: Are there derivatives you can't do?

A: Yes, if they involve functions beyond freshman calculus, like Bessel or other advanced functions. But calc101.com can do any freshman derivative, up to the limits of the computer.

### Q: Are there integrals you can't do?

A: Again, calc101.com can only do freshman integrals. Out of nearly four thousand tests from four calculus books, there were a few calc101.com couldn't do. That's better than 99.9%.

To avoid bogging down the system with huge problems, a maximum of 50 steps are allowed and there is a time limit for each integral. You should be able to get the solution to almost any integral from a standard calculus book without a problem.

If you don't get an answer to an integral that came from a calculus book, check your input carefully to make sure you are following the right input format.

### Q: Why can't calc101 integrate any function?

It is easy to type in simple-looking integrals that cannot be done by anyone, anywhere. Probably the random example x^2 tan[x+1/x] is like that. You are never charged for a tough integral that calc101.com can't do.

Over time mathematicians have figured out how to solve many classes of advanced integrals, but only by going far beyond the likes of trig or hyperbolic functions.

### Q: Do you have any examples of advanced integrals calc101 can't do?

A: Sure. These functions cannot be integrated in terms of elementary functions: e^-x^2, sin[x]/x, cos[x]/x, 1/ln[x]. In fact their integrals are used to define new advanced functions.

### Q: Can I do implicit differentiation?

A: Sure; use y[x] for a function of x. For example, input x^2 + y[x]^2. (If you only enter x^2 + y^2, y will be treated as a constant so its derivative will be zero.)

### Q: Can I do definite integrals?

Not directly, sorry. First find the indefinite integral using calc101, then substitute the upper and lower limits of integration by hand and subtract. This will work if the function is continuous on the interval. Unfortunately checking that a function is continuous is very hard to program, as is finding the limit of a function.

### Q: How do I know I'm getting the right answer?

A: For a derivative the answer will be right 100% of the time, guaranteed.

For an integral, calc101.com does a careful internal check before showing a solution. If it is wrong you will get a message saying that calc101.com could not do the integral. Your account will not be debited unless the solution is correct.

Very rarely, in spite of many checks, it may happen that calc101.com gives a wrong answer for an integral. If you notice an example, please send email!

Of course, if you input x sin[x]^2 but you meant x sin[x^2], you will get an answer and your account will be debited. calc101.com can't tell the difference between what you mean and what you type.

### Q: What are the most common input mistakes?

A: Currently the most common input mistake is forgetting brackets for compositions of functions, like sin[sqrt(x)] (wrong) instead of sin[sqrt[x]] (right). Your account will not be debited for mistakes like these, but they slow you down.

### Q: Am I allowed to store my answers?

A: Sure. Print your results or save them to your machine's hard drive for future reference.

### Q: Sometimes I don't get any response at all even after trying a couple of times. Why?

A: If the answer is too big, your browser may have trouble displaying it. Try a variation of the integral by lowering a power, for example.

Another possibility is that some part of the system is down. Wait a while before trying again. It's good that you tried more than once, though.

### Q: Why don't you show the steps for integrating sec[x]?

A: The integrals of sec[x] and a few other functions come up almost as often as the integrals of sin[x] and cos[x]. They are derived by tricks that take a few steps. So these special trig cases are separated out to avoid cluttering solutions.

### Q: Why don't you show the steps for reduction formulas?

A: The derivations of reduction formulas are also shown separately. To derive a reduction formula you usually need to do integration by parts twice and then solve for the original integral with a bit of algebra. calc101.com simply applies a reduction formula the way people usually do.

### Q: Why don't you show the steps for breaking up a fraction into partial fractions? You only show the result.

A: You can get partial fraction decomposition at partial fractions for free. The program was written by guest programmer Sam Blake.

### Q: Can you integrate a function that has a symbolic parameter, like sin[a x]?

A: Sure. calc101.com can integrate sin[a x] and many other similar cases. By the way, make sure you separate the parameter from the variable by a space or an asterisk * to indicate multiplication: sin[a x] and sin[a*x] will work but sin[ax] won't work.

### Q: Is there ever a problem with a symbolic parameter?

A: Yes, some integrals with a symbolic parameter will fail. For example, a reduction formula works powers down from n to either n-1 or n-2. For calc101.com to be able to finish, the number n has to be an actual positive integer, not left as a symbolic parameter.

You can try any integral knowing that your account will not be debited if calc101.com can't do it. If an integral doesn't work with a parameter, you can try to find a general pattern by doing several examples with specific values of the parameter.

### Q: Why can't I use u, D, or theta in my function?

A: In principle variables and functions could be anything, but in practice there are very confusing choices and some technical limitations. Therefore not all letters are allowed.

Multiplication is indicated by a space or an asterisk: b sin[x] or b*sin[x]. However it is a very common mistake to run variables and functions together incorrectly, like bsin[x] (wrong!). This kind of bad input can lead to a lot of confusion!

The solution adopted in calc101 is to only allow two kinds of functions. First, the standard elementary functions like sin[x], log[x], and sqrt[x] are allowed. Second, calc101 can also handle symbolic unknown functions, so expressions like f[x], y[t], and g[z] are also allowed.

That blocks out typical typing mistakes like bsin[x], so you won't be charged for something that you probably don't want.

Parameters are also restricted to be a single letter. So b sin[c x] is fine, but mu sin[theta] is not allowed--use M sin[t] instead. The number pi (3.14159...) is allowed, though.

You can index parameters; for example, a[2] and k[3] are fine.

### Q: What letters can I use?

A: You can use these letters as the variable of integration, parameters or function names:
a, b, c, -, e, f, g, h, i, j, k, l, -, -, o, p, q, r, s, t, -, -, w, x, y, z,
A, B, -, -, E, F, G, H, -, J, K, L, M, -, -, P, Q, R, S, T, U, V, W, X, Y, Z
.

exception: The letter e is the base of the natural logarithms (2.71828...). You can use e in functions, like x e^x^2, but not as the variable of integration in the "with respect to" field, or as the name of an unknown function.

The number pi (3.14159...) is similar.

### Q: What letters can't I use?

A: Avoid d, m, n, u, v, which are used by calc101 and C, D, I, N, O which have a meaning in Mathematica.

#### in calc101.com

• d is used in derivatives and integrals as the differential.
• m and n are used to explain integrals of trig products.
• u and v are used to explain the product, quotient and chain rules.

#### in Mathematica

• C is used for indexed constants of integration when solving differential equations.
• D means the derivative operator.
• I is the square root of minus 1.
• N is the numerical function that turns symbolic numbers like sqrt[2] into decimals like 1.41...
• O is used in series as the big-oh notation.

### Q: Why are the answers I get sometimes different from my calculus book?

A: Even though calc101.com does the steps to follow what a human would usually do, it may not solve a problem exactly like your textbook or teacher. That's because there may be a choice of different methods at various points. If the problem is simple enough, the answers will usually look the same.

### Q: My text book uses absolute values in the answers for the derivative and integral of arcsec[x], but you don't. Why?

A: calc101.com makes the valid assumption that variables are in domains that allow absolute values to be ignored. This keeps the work simpler.

This assumption is legitimate, according to Remark 1, p. 239, Calculus and Analytic Geometry, Fourth Edition, by George B. Thomas Jr., published by Addison-Wesley, 1968. We won't go into the details here. Essentially the idea is to define the principal values of these functions for negative values of x to be in appropriate intervals. By the way, the functions arccsc[x] and arcsech[x] are similar to arcsec[x]. Of course, none of this matters if x is positive.

### Q: Speaking of calculus books, why don't you explain calculus concepts like they do?

A: This site supplements text books by giving you step-by-step solutions to the two basic operations of calculus. There are hundreds of books and web sites that explain calculus. Calc101.com does calculus.

### Q: What is that funny looking e?

A: The one that has an extra line through it? That's e (approximately 2.71828...), the base of the natural logarithms. It is displayed that way in output. You input it as e.

### Q: I got a really weird result for an integral. What happened?

A: The most likely reason is that you typed in something that you didn't intend. For instance, if you enter x / x^2 + 1 when you mean x / (x^2 + 1), it might look like calc101.com made a silly mistake.

calc101.com will process any input that is in the correct syntax. It doesn't know you meant something else. Not using the right input format, mistyping, or miscopying from your book may lead to puzzling results.

### Q: How does calc101.com work?

A: calc101.com uses a system of completely automatic computer programs. It doesn't need any human intervention to work. That means you can get an answer any time of the day, seven days a week.

The program that takes derivatives has 15 rules. They more or less correspond to the usual rules you learn in calculus, like the product rule and the chain rule. The program that does integration is much more complex, with more than 300 rules. For integrals, the main problem was to arrange the rules so that the right one worked at the right time.

The step-by-step programs in calc101.com are written in Mathematica, a software system with thousands of mathematical, graphics, numerical, and programming functions. Mathematica includes derivatives and integrals, not to mention solvers for algebraic and differential equations, and much more.

### Q: How was calc101.com designed?

A: To make solutions clear, natural, and as easy to follow as possible. Here's how.

• In a sum of integrals, calc101.com tells you which part it is working on. You don't have to hunt for the change from one line to the next.
• Because each step is specifically explained, you don't have to puzzle over what changed from one step to the next.
• If the same substitution can be used more than once, calc101.com will remember. That saves steps when substituting back at the end.
• The hardest part of an integral gets done first. This avoids repeating steps, especially when using a reduction formula.
• If it makes sense, calc101.com factors the final answer, giving you another way to look at the result.