Calculus Errors

Many of the errors listed here are not really calculus errors, but errors that commonly occur in a calculus class and notational errors that are calculus related.  If you haven’t had a calculus class then I would suggest that you not bother with this section as it probably won’t make a lot of sense to you.

If you are just starting a calculus class then I would also suggest that you be very careful with reading this.  At some level this part is intended to be read by a student taking a calculus course as he/she is taking the course.  In other words, after you’ve covered limits come back and look at the issues involving limits, then do the same after you’ve covered derivatives and then with integrals.  Do not read this prior to the class and try to figure out how calculus works based on the few examples that I’ve given here!  This will only cause you a great amount of grief down the road.

Derivatives and Integrals of Products/Quotients

Recall that while

are true, the same thing can’t be done for products and quotients.  In other words,

If you need convincing of this consider the example of   and .

I only did the case of the derivative of a product, but clearly the two aren’t equal!  I’ll leave it to you to check the remaining three cases if you’d like to.

Remember that in the case of derivatives we’ve got the product and quotient rule. In the case of integrals there are no such rules and when faced with an integral of a product or quotient they will have to be dealt with on a case by case basis.

Proper use of the formula for  

Many students forget that there is a restriction on this integration formula, so for the record here is the formula along with the restriction.

That restriction is incredibly important because if we allowed  we would get division by zero in the formula!  Here is what I see far too many students do when faced with this integral.

THIS ISN’T TRUE!!!!!!  There are all sorts of problems with this.  First there’s the improper use of the formula, then there is the division by zero problem!  This should NEVER be done this way.

Recall that the correct integral of  is,

This leads us to the next error.

Dropping the absolute value when integrating  

Recall that in the formula

the absolute value bars on the argument are required!  It is certainly true that on occasion they can be dropped after the integration is done, but they are required in most cases.  For instance contrast the two integrals,

In the first case the  is positive and adding 10 on will not change that fact so since  we can drop the absolute value bars.  In the second case however, since we don’t what the value of x is, there is no way to know the sign of  and so the absolute value bars are required.

Improper use of the formula  

Gotten the impression yet that there are more than a few mistakes made by students when integrating ?   I hope so, because many students loose huge amounts of points on these mistakes.  This is the last one that I’ll be covering however.

In this case, students seem to make the mistake of assuming that if  integrates to  then so must one over anything!  The following table gives some examples of incorrect uses of this formula.

Integral	Incorrect Answer	Correct Answer

So, be careful when attempting to use this formula.  This formula can only be used when the integral is of the form  .  Often, an integral can be written in this form with an appropriate u-substitution (the two integrals from previous example for instance), but if it can’t be then the integral will NOT use this formula so don’t try to.

Improper use of Integration formulas in general

This one is really the same issue as the previous one, but so many students have trouble with logarithms that I wanted to treat that example separately to make the point.

So, as with the previous issue students tend to try and use “simple” formulas that they know to be true on integrals that, on the surface, kind of look the same.  So, for instance we’ve got the following two formulas,

The mistake here is to assume that if these are true then the following must also be true.

This just isn’t true!  The first set of formulas work because it is the square root of a single variable or a single variable squared.  If there is anything other than a single u under the square root or being squared then those formulas are worthless.  On occasion these will hold for things other than a single u, but in general they won’t hold so be careful!

Here’s another table with a couple of examples of these formulas not being used correctly.

Integral	Incorrect Answer	Correct Answer

If you aren’t convinced that the incorrect answers really aren’t correct then remember that you can always check you answers to indefinite integrals by differentiating the answer.  If you did everything correctly you should get the function you originally integrated, although in each case it will take some simplification to get the answers to be the same.

Also, if you don’t see how to get the correct answer for these they typically show up in a Calculus II class.  The second however, you could do with only Calculus I under your belt if you can remember an appropriate trig formula.

Dropping limit notation

The remainder of the errors in this document consists mostly of notational errors that students tend to make. 

I’ll start with limits.  Students tend to get lazy and start dropping limit notation after the first step.  For example, an incorrectly worked problem is

There are several things wrong with this.  First, when you drop the limit symbol you are saying that you’ve in fact taken the limit.  So, in the first equality,

you are saying that the value of the limit is

and this is clearly not the case.  Also, in the final equality,

you are making the claim that each side is the same, but this is only true provided   and what you really are trying to say is

You may know what you mean, but someone else will have a very hard time deciphering your work.  Also, your instructor will not know what you mean by this and won’t know if you understand that the limit symbols are required in every step until you actually take the limit.  If you are one of my students, I won’t even try to read your mind and I will assume that you didn’t understand and take points off accordingly.

So, while you may feel that it is silly and unnecessary to write limits down at every step it is proper notation and in my class I expect you to use proper notation.  The correct way to work this limit is.

The limit is required at every step until you actually take the limit, at which point the limit must be dropped as I have done above.

Improper derivative notation

When asked to differentiate  I will get the following for an answer on occasion.

This is again a situation where you may know what you’re intending to say here, but anyone else who reads this will come away with the idea that  and that is clearly NOT what you are trying to say.  However, it IS what you are saying when you write it this way.

The proper notation is

Loss of integration notation

There are many dropped notation errors that occur with integrals.  Let’s start with this example.

As with the derivative example above, both of these equalities are incorrect.  The minute you drop the integral sign you are saying that you’ve done the integral!  So, this means that the first equality is saying that the value of the integral is , when in reality all you’re doing is simplifying the function.  Likewise, the last equality says that the two functions,  and  are equal, when they are not!  Here is the correct way to work this problem.