# How can you be perfect at maths (highschool)?

I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semester:

• Limit of a function at a point
• Limits Theorems
• Limits of fractional functions
• Limits of Trigonometric functions
• Limits at Infinity
• Continuity at a point
• Continuity on an interval
• Rate of Change
• First derivative
• Continuity and differentiation
• Differentiation Rules
• Derivatives of Higher Order
• The chain rule
• Implicit differentiation
• Geometric applications of differentiation
• Physical applications of differentiation
• Related Rates
• Increasing and Decreasing functions
• Extreme Values.

Limits are relatively easy. However, related rates and extreme values are disgustingly difficult, is there any way to make those two lessons easy and routine? Something like a book filled with questions on those two or something.

Thanks.

• I've retagged this <calculus> instead of <differential-equations> and added the <secondary-education> tag, although it must be said that the topics mentioned are often also covered in early undergraduate calculus. – J W Oct 17 '14 at 13:52
• I would be really interested to hear why you are aiming for the perfect grade. – Andrej Bauer Oct 17 '14 at 21:21
• A challenge is good, but focusing on learning and enjoying are good goals too... – copper.hat Oct 18 '14 at 17:12
• One very helpful approach to getting the best possible grade is to cease thinking in terms of getting the best possible grade, and to entirely embrace the subject itself. Understanding the myriad small details is much easier when they are not "drilled" as separate or unrelated details, but as intensely interconnected manifestations of a few larger ideas. – paul garrett Oct 18 '14 at 21:26

The best way to ensure a good grade is to make sure you deeply understand the topics you are supposed to learn. It is of course important to remember the routine solution methods, but you should also be able to tell intuitively and at a glance why these methods work and where any given method is applicable. You of course need to remember some key results, but you should also be able to justify those results — or even better, give a (sketch of a) proof.

The point is that if you understand the topic well, you can quickly and reliably reconstruct all necessary information. If you remember the topic as a whole, it does not matter if you forget some little details. I have a PhD in mathematics and I still occasionally forget elementary things, but I can fill the gaps. For example, if you remember the differentiation rule of quotients but you are not sure about the signs, test it with some simple functions — the sign in the general case must be the same as in any example.

Teachers often focus on telling what is true (differentiation rules, ways to calculate limits), but I strongly recommend learning also what is not true. For example, if $\lim_{x\to\infty}f(x)=0$ and $\lim_{x\to\infty}g(x)=\infty$, do we necessarily have $0<\lim_{x\to\infty}f(x)g(x)<\infty$? If you are aware of some common "false rules" that are easy to believe, you can recognize when you have made a mistake. When solving a problem, try to make sure that you understand what you are doing at all times and test your claims in special cases if you are unsure. (The last sentence may sound trivial, but many students seem not to do this.)

You will make mistakes and you will forget things. We all do. If you want to make yourself good, try to make yourself robust — so that if you forget something, you can reconstruct it based on something else, and if you make a mistake, you can recognize it yourself.

So far I have answered a question like this: "What kind of a student will almost surely get perfect grades?" Another important question is: "How does one become such a student?"

For one thing, you should know what you want to become. If you really want to understand mathematics well, examine your own skills. Ask yourself what are the most important ideas, results and methods in higher order differentiation. If you cannot answer with confidence and give a couple of examples demonstrating these ideas, you need to work more.

For another thing, do not limit your scope to the present course if possible. The big picture you create for yourself shouldn't be only about the course at hand, but mathematics as a whole. I would even suggest not trying to remember which course a given topic was covered in and which course you are having at the moment. The borders between different courses are somewhat artificial and you don't need to respect them.

Also, if you have the extra time, look what is coming ahead: find a follow-up course that builds on your current course and take a look at its book. When I was in high school (or the closest equivalent in Finland), other students thought that I didn't have to work at all because I understood quickly and could solve problems quite intuitively. The reason was that I was working ahead of them: I had already read the book of the next course, and that gave me plenty of context and motivation for the present topic and I could focus on building a solid big picture. I was working hard, but I was working on something different than others. It often happens that you properly understand something only when you have applied it in something else; no one masters the last thing they have learned.

As JPBurke suggests, working in a group also helps. But a group is not strictly necessary if you can't find equally motivated friends or suitable ways to collaborate. What you do need is someone to ask from if you don't understand something on your own. It can be a fellow student, a teacher, an older sibling or anyone willing to help.

I realize that this answer gives somewhat grandiose goals. A perfect understanding is too much to ask for, but I do suggest putting goals in this direction. For me playful interest and idle curiosity in mathematics is what kept and still keeps me going; there is no need to be serious in order to become good. The most valuable thing you can have when trying to get good grades is a passion to understand.

• I have found that reading the book about 3 weeks ahead of time is very useful. Reading ahead by a full term is also useful, but 3 weeks is a good head-start. It is very helpful when taking physics classes to have already taken the corresponding calculus class. – Jasper Oct 17 '14 at 21:18
• I agree with @Jasper: 3 weeks is already very useful. I would even say that any reading ahead of time is always useful for all students. Even half a week helps, but longer is better. It is easier to absorb the main points when not everything you hear in class is new. – Joonas Ilmavirta Oct 17 '14 at 21:26
• Great answer. I especially liked this part: "You will make mistakes and you will forget things. We all do. If you want to make yourself good, try to make yourself robust ‐ so that if you forget something, you can reconstruct it based on something else, and if you make a mistake, you can recognize it yourself." – Michael Joyce Mar 16 '15 at 14:14

In college I had a friend who was also a Math major, and had been in the Marine Corp. His advice was a little coarse and echoed his military background: "Work problems until you puke, then wipe off the puke and work some more". I followed this advice and was an A student in college. I found that in Calculus in particular, the difficulties I had were due to my weakness in algebra and trigonometry. The equations in Calculus problems will almost always simplify. Hope this helps!

• yes, do more problems! – rbp Oct 17 '14 at 20:45
• That's a good point - all of my students' problems in Calculus aren't Calculus, but the algebra needed to simplify. – Jhecht Mar 15 '15 at 20:36
• Maybe without the necessity of puking, I'd agree that nearly all students' perceived "difficulties" in calculus are actually incapacities in middle-school algebra, ... not to mention trigonometry, although the latter is perverse, in itself, since the way it's usually taught ignores complex numbers and the ultimate clarification of trig functions by observing that they're expressible in terms of the complex exponential... – paul garrett Mar 16 '15 at 0:35
• To be clear: puking is not necessary, but perhaps letting of preconceptions is. :) – paul garrett Mar 16 '15 at 0:36

Welcome to the site!

It's great that you're motivated and want to get a perfect grade in your studies! There is no actual formula for getting a perfect grade. One goal among all the goals of your math educators is to deepen your understanding of the mathematics you are studying in class and in whatever books your reading. There are things you can do to address this as well. I will suggest one. Because you seem so motivated, I think this is something that could work for you.

If you haven't already done this, consider forming a study group with other students in your class. It doesn't matter so much whether they are more or less advanced students than you. A mix may actually be good. The aim, primarily, is to gather a few like-minded people who are motivated to think about mathematics together. It's important that you're comfortable talking together about ideas, and especially putting forward suggestions that may not be "correct."

If you can form such a group (even maybe just 2 or 3 people), have meetings around a number of goals and questions such as:

• Discuss the ideas in a chapter (or chapter section) you all have agreed to read.
• What was challenging about it?
• What seemed obvious about it?
• What are you still unclear about?
• Where might it be useful?
• Did anything seem especially interesting? Exciting? Pointless? Annoying? If so, why?
• Do you feel there is a better way to do something presented in the chapter? Why is this other way better?
• Find online sources of information related to things you've already read in the book chapters.
• Are there differences in how these concepts are presented?
• Did anyone find a presentation they found easier to understand? How does that explanation relate to what you read in your textbook?
• Solve problems you've found in other books (or online)
• What different approaches occur to you?
• What is surprising about these problems?
• Are there problems you just can't solve? Do people get hung up in the same place? Consider asking a teacher if there is some approach that will help you solve this problem.
• Brainstorm applications for the mathematics you're learning.
• What contexts do the textbooks and other sources suggest?
• Where is this mathematics generally used?
• Search online to find what applications the math may relate to. What did different people find? Does it intersect with any of your interests?
• Is the math you're learning part of a trajectory leading to some advanced math you know you will be encountering int he future?

You can come up with your own ways of discussing the mathematics you're learning, I'm sure. Obviously, your group could mostly be about discussing and comparing approaches to solutions, or even working problems collaboratively. The point is that there can be a lot to talk about related to your mathematics learning, so a group like this has very few limits. And these discussions could be very productive.

One word of caution: be very mindful of your teacher's rules about homework collaboration. There's nothing wrong with taking on extra problems and working them collaboratively, but your teacher may have good reasons to establish very specific rules on the homework you've been individually assigned. Just be clear on those rules and you'll be fine.

I feel that the excellent answers given already may not do enough to discourage the idea that you need to be perfect. While I do understand your aim towards a perfect grade, you need to embrace the possibility that this may not happen and to not let that deter you from continuing your math education with the same passion you've shown so far. Grades are an arbitrary measure, which have their place in education, but they must not become what defines you; you are not your grade. It is from failure that we build the most astounding successes. Mathematical perfection in a student is beautiful in the way the smell of a flower makes the tree beautiful: will you reject the refreshing taste of its fruit, the intricate pattern of its leaves, the impressive strength of its root because its flower didn't smell how you expected?

Embrace rigor, and never settle for less than your best, but seeking perfection during your mathematical exploration above all else may cloud the wonderful growth it may spur in you whenever you meet failure, the same failure even your teachers met, as they were preparing to guide you through the wonderful journey ahead.

• Good answer! I should add that insisting on perfection can keep you from moving forward. If you don't get something, move on, and maybe you'll figure it out later. – Joonas Ilmavirta Jul 6 '17 at 9:09

The advice that JPBurke and Joonas Ilmavirta have given is excellent.

If you want to be perfect, you need to check your work (as explained at the end of this answer). You also need to know that some problems do not have answers -- and that the correct answer may be to point out why. Furthermore, real-world problems have limited precision (or accuracy) of the input data. Understand significant figures, and when "close enough" really is "good enough".

With regard to calculus:

• The Fundamental Theorem of Calculus and the Reynolds Transport Theorem are the two most important concepts in calculus. Yes, they are even more important than limits or the definition of a derivative. This is because they let you sanity-check your work in real life, even if you only have estimates available.
• The Fundamental Theorem of Calculus proves that integration and differentiation are inverse processes. You can use integration to check your derivatives, and vice versa.
• The Reynolds Transport Theorem is a detailed version of "what goes in, either stays in, or comes out." It is the basis for all Conservation Law problems, such as almost all practical problems in physics and engineering. Learn this Theorem, and you will have a much easier time in Physics, Quantitative Chemistry, Statics, Mechanics, Electrodynamics, Fluid Mechanics, and Thermodynamics. This is because almost all of the problems in all of these subjects use the same math -- special cases of the Reynolds Transport Theorem.
• Try to solve most problems without using a calculator.
• Follow a good process for solving (and checking) problems, as discussed below.
• For every problem (that is not a proof), graph the problem (either on paper, or in your head). Mark on the graph where the function crosses the x-axis, where it crosses the y-axis, where it has minimum(s) and maximum(s), and where it has inflection points. Know where the slope is positive, zero, or negative. Doing this will seem like a lot of work at first, but it will give you an intuitive sense for the shape of the graph.
• You will learn some important things from the graph practice I just mentioned. The slope of a function is the function's derivative. If the slope is zero, the graph is either at a minimum, a maximum, or an inflection point. Where the second derivative is zero, the graph is probably at an inflection point.
• Pay attention to symmetry. Is the function even or odd? (In other words, is it symmetric with respect to x = 0?) Unshifted even-powered polynomials and unshifted cosines are even; unshifted odd-powered polynomials and unshifted sines are odd.

When you are doing things with vectors, the following notations work well:

• If you are allowed to choose your notation, use x, y, and z's "hat notation" to indicate the directions of unit vectors. For example, "y-hat" is a letter "y" with a circumflex accent, and is written "ŷ". This is less confusing than using the i, j, and k "hat notation" (such as "ĵ").

• Either use angle bracket notation for vectors (such as <x,y,z>) or explicitly add the vectors (such as x x-hat + y ŷ + z ẑ, and use the proper symbol instead of x-hat).

• Unfortunately, you cannot use the hat notation to indicate unit vectors in Quantum Mechanics, because Quantum Mechanics uses hats to indicate "multiplication" by an operator. For example, "ŷ" in Quantum Mechanics means "multiply by y".

With regard to solving word-problems in general:

1. Try to draw a picture.
2. If the problem uses units, keep the units with the associated numbers. For example, if t = 2 seconds, never abbreviate this as t = 2.
3. Label what you know. Label where each variable is zero. Label which direction it is increasing in.
4. Label what you are trying to find.
5. Write out your variables and parameters. For example, "t = time since rocket was fired."
6. Write out the relevant formulas. Count up your knowns and unknowns. For each unknown you have, you need another independent equation if you are to determine the unknown's value. Make sure to include both "boundary conditions" and "field conditions".
7. Systematically solve the equations using algebra and/or calculus. Make sure that you either do not divide by zero, or that you only divide by zero as part of a limit. (Derivatives and L'Hopital's rule use limits to perform "valid" divisions by zero.) Make sure that you bifurcate the problem as necessary. For example, when you factor a polynomial to find its roots, you will have a separate subproblem for each factor.
8. Only after you have found a formula for your answer should you compute a numerical answer. Make sure to write out the units as you compute the numerical answer.
9. Check whether each solution is valid. For example, if you are finding positive solution(s), explicitly rule out any negative solution(s) you find. Also, check whether the units make sense. For example, if the answer is supposed to be in meters per second per second, but your answer is in meters per second, then you have probably made a mistake. If necessary, use unit conversion factors. (For example, multiply a parameter by (1000 ms / s) if the parameter is in seconds, but the answer needs to be in milliseconds.)
10. Clearly label your answer, and make a note about what the answer means. For example, "t = 4 s. The rocket reaches maximum altitude 4 seconds after it is launched." I was taught to circle this answer in a cloud.
11. Sanity check your answer. If the rocket was supposed to go to the moon, 4 seconds seems awfully short.
12. Check-By-Substitution (CBS). Plug your answer into the original formulas (making sure to keep the units). Reduce both sides of each equation until you can either confirm that the answer is (a) correct answer, or that it is incorrect. Mark the initial and intermediate equals-signs with question marks. If the CBS works, mark the final equals-sign with a check mark. If the CBS fails, use an inequality sign, and look for a mistake.

Solve problems.

That's literally the only way to get better in math. Do more problems. Don't do the same problem with different numbers, do harder and harder problems until you are mentally exhausted, take a break, and delve in again.

The best way to do well in math is to solve problems. (Not to deeeeply understand things.) Deeply understanding things is fine. Even beneficial. but skill in problems is more important.

Per the actual question: yes, Schaum's Outlines. Perfect for what you want which is to drill the A.

Oh...and maybe it is just me, but I find myself realizing things about the concepts AS I DRILL.