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the applications problems have vanished; their solutions are present however.

Using \input{subfile}, you can break up a TeX project into smaller bits, which makes editing and general maintenance easier. For a very in-depth look at how to manage TeX projects in a modular way, perhaps see this blog post by yours truly.

I wouldn't recommend using all the advanced stuff in that post, but at least there are complete working examples showing various levels of sophistication. You could start with just the first level.

Appendix is a mess; format and neaten up.

can be changed from economics to a different field

It is strange to me to introduce integration and differentiation before the notion of limits - since both integrals and derivatives are limits, of the Riemann sum and the difference quotient, respectively.

I would propose to move the order around, and introduce first limits, then differentiation, then integration, since differentiation just requires the limit of a quotient, but integration that of a sum/series, which is a slightly more involved concept. Discussing limits first would also allow you to give at least partial formal definitions of differentiation and integration instead of just drawing a picture and then giving a table of fixed results the reader may use to compute some expressions with by using linearity and product rules. After all, while the rules for computing things are important to get result, the thing to understand in calculus is why the derivative of x^n is n x^{n-1}, for instance, not just knowing it is.

Here it is stated that we can break up sums inside of integrals.

Note that this is generally true:

\int f(x) + g(x) = \int f(x) + \int g(x)

One way to think about this is that the act of integration is linear. Think of the indefinite integral as a function that maps {functions of a real variable} to {functions of a real variable}. For example, the indefinite integral maps x^2 to (1/3)x^3. So, as a function, integration is linear because the integral of a sum of functions is equal to the sum of the integral of those functions.

General:

• There is no reason not to use the hyperref package to make the TOC into active links (and all references within the document, too, although currently there aren't any).

• The "resources" section should be a bibliography in the standard LaTeX sense, not its own section

• Your quotation marks are "wrong" - opening quotation marks should be inverted, which is done by using a double backtick instead of the quotation mark (most editors should have an option to do this intelligently for you).

• The d in the derivatives/integral should be upright \mathrm{d}, since it is not a variable.

• Within displaystyle (i.e. displayed equation as in \begin{equation}...end{equation}) the \limits is completely unnecessary and can safely be omitted

Specific:

• Line 42 ff.: The functions and variables should be inline TeX, not standard text.

• Line 68: Riemann, not Rieman

Derivatives

1. Line 1 "Differential calculus is all about finding the slope of a curve. "
2. Line 15 is very inaccurate.
3. Line 18 is not concise.
4. In Line 46 you should mention that c is a constant and n is non-zero.
5. Learning differentiation without learning limit?
6. Use "\mathrm dx" instead of "dx" when typesetting.
7. Line 70 should be in the beginning.
8. Line 64 can be elaborated like each of the three rules below.
9. Line 82 contains no mention that sin/cos = tan.
10. Line 87 is wrong. 1/cos^2(x) does not evaluate to sin^2(x).
    Integrals
11. Line 1 "The point of the integral is to find the area under a curve. "
12. Line 2 area > signed area
13. Line 62 this is not called the product rule
14. Line 63 what is wrong here?
    Limits
15. Line 1 "The purpose of limits is to find what a function approaches at a certain number that it is not defined for. "
16. Line 8 The constant e is not 2.72.
17. Line 23 it's called an "indeterminate form".
18. Line 104 integral?
    Terms To Know
19. Line 16...
20. Line 20 is plain wrong.
21. Line 25: actually you don't do work when carrying a book horizontally.
    Solutions
22. Line 7: use (x^2 - 5) instead of x^2 - 5.
23. Line 10: I think you meant \int^5_3.
24. Line 11: parentheses.
25. Line 18: wrong variable.
26. Line 32: some typo.
27. Line 55: The 5 disappears...
28. Line 57: Hooke's law says F = -kx where k is positive, so you can't have 10x....

Thanks to a user on MSE for reviewing this and giving these comments.