# Calculus

## What is Calculus?

Technically, calculus is the study of rates of change. However, if you’ve never taken calculus before, “rates of change” might have too much meaning to you. Calculus is actually two separate categories: differentiation and integration. Differentiation is a tool where you can find an object’s velocity and acceleration based on the formula for that object’s position. Likewise, if you know the object’s velocity you can find that object’s acceleration. With integration, the opposite is true: you can find an object’s position if you know the object’s velocity or acceleration. This is illustrated in the following diagram.

The branch of calculus that deals with differentiation is called differential calculus and the branch of calculus that deals with integration is called integral calculus.

You probably already have a good idea of what calculus is about if you’ve studied algebra. In algebra, you found the slope of a line using the slope formula, slope = rise/run. In calculus, you’ll be studying the slope of a curve. The slope of a curve isn’t as easy to calculate as the slope of a line, because the slope is different at every point of the curve (and there are technically an infinite amount of points on the curve!).

That’s where differentiation comes in: differentiation is a set of tools that allows you to find the slope of a tangent line at any point on any curve. This slope is called a derivative.

You can calculate the slope of a line at any point on a line by using two points (a and b on the top left picture) and the slope formula. However, you can’t use the same formula to calculate the slope of a point on a curve. Points a and b on the top right picture shows that the two points have very different tangent lines (shown in red). In order to calculate the slope of the tangent line at these points, you need calculus.

## Trigonometric Identities

You’re going to need to be familiar with trigonometric identities (or at least know where to look for them). Trigonometry is an entire semester-long class (sometimes two!), so it isn’t possible to put all of the identities here. But some identities show up a lot more frequently than others. These are the trigonometric identities you’ll use over and over again.

### Reciprocal Identities

What this is telling you is that the reciprocal functions sec, csc and cot are the reciprocals of the cosine, sine and tangent functions.

### Tangent and Cotangent Identities

These identities are telling you that the tan of a function is a ratio of sine and cosine and the cot is the ratio of cos and sin.

### Other Trigonometric Identities

The first identity is really just a version of the Pythagorean theorem.

### What are the Trigonometric Identities Used for in Calculus?

The above trigonometric identities will be used over and over again in your classes. They especially come in handy when it comes to figuring out derivatives or simplifying functions. For example, if you have a sin2 and a cos2 close to each other in a function, you might be able to cancel them out using a trigonometric identity. If you have a messy looking function with sin/cos/-cos2/sec and other components, look for ways to convert to sin or cos using the above trigonometric identities.

### Other trigonometric functions

There are dozens of other possible trigonometric functions like arccosine, arctangent and arcsine, but the reality is you’ll rarely, or never use them. In the five semesters of calc I took in college (calc I/calc 2/advanced calc I/advanced calc II and one semester of calc-based statistics), I think I used the rarer trigonometric functions only once or twice. You’ll probably never see them on a test (they might be assigned as a “tricky” homework problem). If they are on a test, your instructor will (or should) provide a list of lesser-known trigonometric identities. As long as you know (and can use) the above identities, you should be all set for your class.

## Series Convergence Tests

Often, you’ll want to know whether a series converges (i.e. reaches a certain number) or diverges (does not converge). Figuring this out can be an extremely difficult task — something that’s beyond the scope of even a calculus II course. Thankfully, mathematicians before you have calculated the convergence or divergence of many common series. This enables you to figure out whether a particular series may or may not converge.

## Series Convergence Tests in Alphabetical Order

Absolute Convergence
If the absolute value of the series converges, then the series converges.

Alternating Series Convergence Tests
If for all n,an is positive, non-increasing (i.e. 0<=an) and approaches 0, then the following alternating series converges:

If the series converges, then the remainder R,sub>N = S-SN is bounded by |R N|<=aN+1. S is the exact sum of the infinite series and SN is the sum of the first N terms of the series.

Deleting the first N Terms
The following series either both converge or both diverge if N is a positive integer.

Direct Comparison Test
In the direct comparison test, the following two rules apply if 0 <=an <=bn for all n greater than some positive integer N.

Geometric Series Convergence

With the geometric series, if r is between -1 and 1 then the series converges to 1(1-r).

Integral Series Convergence Test
The following series either both converge or both diverge if, for all n>=1, f(n) = an and f is positive, continuous and decreasing. If the series does converge, then the remainder RN is bounded by

Limit Comparison Test
The limit comparison test states that the following series either both converge or both diverge if lim(N–>∞) (anbn where an,bn>0 and L is positive and finite.

nth-Term Test for Divergence
diverges if the sequence {an} doesn’t converge to 0.

P series

If p > 1, then the p-series converges
If 0 < p < 1 then the series diverges
Ratio Test
The following rules apply if for all n, n≠0. L = lim (n—>∞)|an+1an|.
If L<1, then the series converges.
If L>1, then the series diverges.
If L=1, then the ratio test is inconclusive.

Root Test
Let L=lim(n–>∞)|an|1/n
If <, then the series converges.
If >, then the series diverges.
If L=1, then the ratio test is inconclusive.

Taylor Series Convergence
The Taylor series converges if f has derivatives of all orders on an interval “I” centered at c, if lim(n–>∞)RN=0 for all x in l:

The Taylor series remainder of RN=S-SN is equal to (1/(n+1)!)f(n+1)(z)(x-c)n+1 where z is a constant between x and c.