Calculus Pitfalls

1. Calculus Pitfalls By Jordan D. White

2. Background Pitfalls • There are a lot of common calculus mistakes that one can make as they go through a problem. • However these are not the only kind of mistakes that you can make in calculus. • Far too often students struggle with arithmetic and algebraic errors. Having a strong algebraic background is key. This is true whether you are taking traditional calculus or applied calculus. • Another issue that may come up if you are taking the traditional route in calculus is trig mistakes. • In any case, having a strong background in the aforementioned subject matter is crucial for calculus and other math classes going forward.

3. Products and Quotients (Products for Derivatives) • When you have a sum or a difference and you take the derivative, you have . • It is easy to apply this sort of logic to products, and very tempting as well. • We may think of a product and think the following, . • While this would great, . Instead . • Say we have two functions and . If we take these functions and take the derivative of the product both ways we get different results. • To check which one works better, we can actually take the product, simplify and use the power rule for derivatives. You’ll find the correct way will equal the result.

4. Products and Quotients (Quotients for Derivatives) • When you have a sum or a difference and you take the derivative, we have established that you have . • Much like products, it is tempting to apply this logic to quotients. • We may think of a quotient and think . • While this would be great, . Instead, . • Say we have two functions and . If we take these functions and take the derivative of the quotient both ways we get different results. • To check which one works better, we can actually take the product, simplify and use the power rule for derivatives. You’ll find the right way will equal the result. • It is easy to confuse the quotient rule with L’Hospital’s Rule.

5. Products and Quotients (Products for Integrals) • When you have a sum or a difference and you take the derivative, you have . • It is easy to think this sort of logic to products. Unfortunately it does not. • We may think of a product and think the following,. • While this would great, . Instead we could have something that requires substitution, integration by parts, or other special cases to approach. Integrating a product is not always straightforward. • Say we have two functions and . If we take these functions and integrate the products separately, we get something different than we should. • We’d get a different result than if we simply used the power rule for integrals on the product. • Were we to check and take a the derivative of the incorrect result we would not be able to get that initial product.

6. Products and Quotients (Quotients for Integrals) • When you have a sum or a difference and you take the derivative, we have established that you have . • It is easy to think this sort of logic to quotients. Unfortunately it does not. • We may think of a quotient and think the following,.dx = • While this would be great,dx . Instead we could have something that requires substitution, integration by parts, or other special cases to approach it. Integrating a quotient is not always straightforward. • Say we have two functions and . If we take these functions and integrate the expressions in the quotients separately, we get something different than we should. • We’d get a different result than if we simply used the power rule for integrals. • Were we to check and take a the derivative of the incorrect result we would not be to get that initial quotient.

7. Exponents and Logarithms (Derivatives) • When dealing with powers , if you have , then . When you have a function of the form , it can be tempting to say the derivative is , however this is not the case. • When you have, , then . • Note, this holds for . We’re told . This is true, it is born from this. We have . • With , then . It follows that with , then We have . • It is easy to forget if you have other functions or numbers involved.

8. Exponents and Logarithms(Integrals) • The power rule for integration is . • It is very tempting to think holds for all n. • However it does not hold for . • Think about what happens when you have that situation. • So instead of using the power for , we have . • Note the absolute value gets dropped a lot, but it is needed. Thinking about ln(x) and what it is, x needs to positive.

9. Constants (Derivatives and Integrals) • In both cases with derivatives and integrals, the constant is along for the ride. • For integration, forgetting the constant of integration happens sometimes. • If you are doing indefinite integration, then we don’t know what the c is and not having it suggests that c is zero, which we can’t know in some situations. • For definite integration, this is not an issue.

10. Other Pitfalls • We already mentioned arithmetic, algebraic, and trig mistakes. However there are other common mistakes. • Notation is a big issue, so be aware of appropriate notation and what it means is important. • Making assumptions about infinity is another thing to consider. You will want to be aware of how it functions and be aware of what you can do with it. • Reading and writing math. Knowing how to read math is key, and knowing how to write math as well. You will want to organized, as though you are writing a paper. • The problem you are approaching is the argument you’re making, it is what you’re trying to show or prove. Math is the language you are writing in. Know your rules for math and have structure to your solutions.

11. Additional Resources • MathWorkshops: https://lrc.umbc.edu/tutor/math-lab/math-workshops/ (How to be a Prime Math Student, Math Anxiety, and Algebraic Pitfalls Workshops). • Calculus Resources: https://lrc.umbc.edu/math-151-resources/ (Notes, Videos, Practice Sets, Links to resources for other classes that preqs for Calculus.) • Math and Science Tutoring Center: https://lrc.umbc.edu/tutor/math-lab/math-lab-courses/#Math151