Reference request regarding calculus exam

Question

I'm currently a first year computer science student and I'm deeply interested in calculus . That being said, what we studied so far consists of: Cantor sets, sequences and a brief introduction to series and convergence tests.

Our final exam is going to be set sometime in February and I took the liberty of looking up exams from past years. Since the structure is mainly the same, I'll post the relevant content of a final:

1. Find the Fourier series for $f(x) = x\left|x\right|$ on $[ - \pi ,\pi ]$ and the value of the series for $x = \pi$.
2. Compute the integral $\int\limits_0^1 {\frac{x}{{\sqrt[3]{{1 - x^3 }}}}}\,\mathrm dx $ using Beta integrals and study its convergence
3. Compute $\int\int_D xy\,\mathrm dx\,\mathrm dy $ where $D = \left\{ (x,y)\mid 2x^2 + y^2 \le 2,x \ge 0,x \le y \le 3x\right\}$.
4. Find the volume bounded by the following two surfaces: $x^2 + y^2 = 1,\,\,z = \sqrt {x^2 + y^2 } ,z \ge 0$ .
5. Approximate the value of this integral to two decimal places: $\int\limits_0^{\frac{1}{4}}\mathrm e^{-x^2}\,\mathrm dx $.
6. Find the extrema of $f(x,y) = 4(x - y) - x^2 - y^2$.

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I know this might be a lot to ask for free, but I really like calculus and I want to learn on my own. I'm really eager to study but I need a bit of guidance. So, to get to the question, can anyone suggest some reading materials that could cover these areas?

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EDIT : I posted this question almost a week ago hoping someone would answer,after seeing that my post didn't get much attention I placed a bounty on it, but still nobody comes forward with answers/comments. Since apparently no one can shed any useful insight on this matter, maybe it is because my post is poorly structured.

If so, can you at least suggest an edit ? Thanks.

Answer 1

I'll give my personal recommendations for each topic, though most of these problems can be solved with basic knowledge of their respective topics.

Mulivariable Calculus: "Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics)" by James L. Callahan.

(That's for more on calculus of several variables and some vector calculus, though for much more on vector calculus [Stokes' Theorem, Green's Theorem, Divergence Theorem, etc.] I recommend searching for something else that's good here on MSE, as I don't know of any)

Fourier Analysis: "Fourier Series (Dover Books on Mathematics)" by Georgi P. Tolstov

Hypergeometric Functions: "Hypergeometric Functions and Their Applications (Texts in Applied Mathematics)" by James B. Seaborn

(For the above, try to get familiar with a bit of [partial] differential equations beforehand if you want a bit of an edge)

In fact, have you tried out studying differential equations?

Answer 2

At my University, I studied these topics from:

Mathematical Methods for Physics and Engineering by KF Riley, MP Hobson and SJ Bence.

You can look this up in google.

However, I must admit that this book might not be a good source of reference for rigorous treatment of calculus. There are plenty of other books that do this. I suggested this book looking at your question.

I hope this helps!