How should I start solving a calculus problem?

Question

There are so many books teaching how to take derivative and integration of a function. I think I'm good enough (enough for me lol) in those parts, my problem is that I can't start solving a question and even don't know where to start (finding or making the correct function or equation, kind of calculus applications in area, volume, etc.) - what should I do?

Plus I wanna know if there is a good syntax in calculus; like if I should write constants and variables in a special way (capital, small, etc.), so that my writing is more readable and clean?

Answer 1

• Understand what derivatives are. Derivatives say something about the rate of change of something; they are not just some algorithm to apply. Try to see the world around you in terms of rates of change and how you might model that mathematically; I think this would give you a better feel for the word problems and how to translate the words into math!
• Read this book: How to Solve it - I'm serious, this is a classic and it teaches you how to think about problems and approach them the right way!
• Review your work! If you've struggled with a particular problem and have finally found the solution (or have been shown it), think about which of your attempts were correct and which were unfruitful, and also think about why you went down the wrong roads. Getting feedback and incorporating it consciously is essential.
• Take your time! Clean writing demands not only that you write with enough space between the lines (and that you write slowly enough to make it look neat), but also that you think about what you are writing and where you are going. Not only will this lead to fewer basic calculation mistakes, but it will also allow you to think about your strategy of solving your particular problem, and this can save you a lot of time in the end: By dividing your problem up into smaller problems and then trying to make a plan on how to tackle each sub-problem, you can easily see if you're going toward a dead end or not. This last thing is also taught in the book mentioned above (read it!).

As for syntax,

• $x,y,z,w$ and so forth (letters from the end of the alphabet) are typically used for variables;
• $a,b,c,d$ (letters from the beginning of the alphabet) are used for coefficients (for instance in polynomials);
• $c$ or $C$ are typically used for constants (sometimes $k$ is used, especially if you're talking about a constant of proportionality); and
• $i,j,k,l,m,n$ are typically used for integers.

Also, when you write derivatives, use the classic $\frac{d}{dx}$ notation if it's concerning an expression (e.g. $\frac{d}{dx}e^{ax}$) but consider using just a prime or a dot if it's regarding a function (e.g. $y'=-ky''$ or $\dot y=-k \ddot y$).
And when you run out of normal letters, there's a whole Greek alphabet. :)

I hope that helps!

Answer 2

To add onto Lovsovs's excellent answer, I encourage you to browse this site (and others) for examples of good mathematical exposition. It may be a little difficult to recognize this at first, but if you look around at problems that are at your level, and find answers that take the time to explain how they proceed to a solution, you will begin, I think, to see that answers have a kind of "shape" to them, as opposed to a bland sequence of steps toward an end goal.