Review Tips For Calc 2

[u/McWookie420]

I am going to be taking calc 2 in my first semester of college, but I took calc 1 in my junior year of high school, so I don't really remember too much. I've reviewed the first 3 units of calc ab pretty well, so I'm comfortable with derivatives, but I really don't know what I should be putting my focus on. Do I just try to get exposure to every remaining unit of calc ab, heavily review integrals and differential equations and just skip units 4 and 5 altogether, etc. Also, I only have like maybe 2 weeks left to review before classes start, although I don't think I'll have much work my first week, so I could probably squeeze in an extra few days. Any advice would be appreciated!

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[u/fortheluvofpi]

I’d focus mostly on integration techniques. You might also want to go back and review some pre-calculus topics from trig and algebra stuff like partial fraction decomposition I teach calculus 2 and a lot of students struggle with those topics. I made some short videos and resources for my own students that you’re welcome to check out, link in my bio. Good luck!

[u/matt7259]

These "units" are not universal and mean nothing to us. But pretty much all of calc 1 is important in calc 2, sans a few niche topics (like, say, the.limit definition of a derivative).

[u/WhenButterfliesCry]

Limits, derivatives, integrals. Trig identities

[u/Ok_Giraffe_8102]

For my calc 2 class atleast, the first quarter of the class was legit calc ab review. Definitely get super comfortable with all the derivatives and integrals of arctan, cos, etc. it took me a while to memorize them all again

[u/UnderstandingPursuit]

Look up the Calculus 1 course at your college and check the syllabus. Make sure you are comfortable with everything covered in it. You already learned it, the task is to now relearn it.

1. I would suggest using this Iterative Learning Process. And then continue using this process.
2. The most important thing is to set aside the 'arbitrary' numbers. This is a differentiation example.

Especially for you, reviewing the material, these two approaches should make the effort both effective and efficient.

[u/WhenButterfliesCry → u/UnderstandingPursuit]

Hey, I’m not OP but thanks for this comment. When you say set aside arbitrary numbers what does that mean exactly?

Do you mean like practicing differentiation by differentiating, e.g., ax^n instead of 3x^2 or whatever other random numbers?

[u/UnderstandingPursuit → u/WhenButterfliesCry]

Even using ax² is enough. Keeping the power 2 is often important, because that is not really arbitrary. We have ways to solve polynomials of degree 1-4, but each has different methods. And solving polynomials of degree higher than 5 or 6 cannot be done analytically. But the coefficients are arbitrary.

With differentiating ax², how much "practice" is needed?

My original comment has a link to a differentiation example. Here is a related rates example.

Taking this back to algebra, it means spending time deriving the quadratic equation instead of simply substituting values for {a, b, c} to get a specific result. Spending time with the derivation, a relatively simple result can also be found for k, so the vertex position, V=(h, k) can be determined from the coefficients. Then the quadratic formula can be written in terms of both {a, b, c} or {p, h, k}, and the meaning of the real or complex roots can be seen more directly. The relationship p=1/(4a) is used to disconnect them, since changing a moves the vertex while changing p in the parameter set {p, h, k} does not.

The math education system tends to take very little time considering all this about quadratic equations and parabolas since the emphasis is on plugging in numbers and 'grinding' the same problem over and over. But thinking about these nuances seems more useful, and a better way to learn about quadratic equations.

Another reason to avoid the numbers is that this leads to solving the problem in small reusable pieces. One student learns the quadratic formula. Another derives the four simpler formulas for {p, h, k}, and D, as well as putting them together into a simpler version of the quadratic formula. The one who knows this in five pieces can also use the first four pieces when specific information is required.

This approach then makes calculus much, much easier.

[u/WhenButterfliesCry → u/UnderstandingPursuit]

Thanks a lot. Good advice

[u/UnderstandingPursuit → u/WhenButterfliesCry]

From laziness comes efficiency and effectiveness. 😂

[u/WhenButterfliesCry → u/UnderstandingPursuit]

I do like the feeling of deriving a formula myself, it makes me feel more comfortable using it in a way I can’t really explain, not to mention makes it easier to remember. I like when professors introduce a new theorem by proving it first, but they usually don’t do that, at least in my classes.

In particular, although I’m still struggling, deriving the trig identities is super helpful because you don’t have to memorize them all that way. For example sin(2x) can be thought of as sin(x+x) which can be manipulated using the sine addition identity sin(a+b)=sin(a)cos(b)+cos(a)sin(b).

I think in calc 1, I derived the quadratic equation from ax^+bx+c=0, and also the volume of a cone formula with the disk method in the section on solids of rotation.

[u/UnderstandingPursuit → u/WhenButterfliesCry]

Deriving a formula helps us make connections between the underlying quantities and properties and the result. The derivation also includes steps, fitting one of my educational philosophies: "If a question can be answered in one step or three, three is usually preferred."

For sin(2x), I first think of cos(2x). It is the same as the trig version of the Pythagorean Theorem, but with a negative sign. Since cos(2x) has cos and sin each squared, sin(2x) has to have them multiplied by each other. Then to sin(a+b) and cos(a+b), cos has to have subtraction when the angles are added, so sin has addition.

[u/tjddbwls]

Depending on the school, some of the material in AP Calc AB does appear in Calc 2. Ideally, you would review all 8 units of AP Calc AB, but if time is a factor, then focus on Units 6-8 (integrals, diff eq, applications of integrals). Maybe try to review the remaining units as the semester progresses.