DON’T use L’Hôpital’s Rule! The function is continuous at \(x=0\).

Just calculate the integral directly. Watch out for the \(2\), and make sure to plug in \(0\) correctly.

Draw a Venn Diagram for \(A\) and \(B\), and reason about how \(B\) must be related to \(A\). (Does one set have to live inside the other?)

Yes, \(|x|\) is differentiable as long as \(x\neq 0\): its derivative is \(-1\) if \(x<0\) and \(1\) if \(x>0\).

You could do a bunch of calculus, but all you need to do is sketch \(y=x^2+x\) (what are its roots?) and look at the graph.

Remember that \(e^x>0\) for all real \(x\).

The expression under the square root needs to be positive. Make sure your intervals don’t contain anywhere that \(\tan ⁡x\) blows up to infinity.

Try a \(u\)-substitution with the denominator.

Don’t bother calculating \(f(x)\) — just try all the solutions.

Calculate \(\dfrac{\partial \phi}{\partial y}\) first.

Use L’Hôpital’s Rule.

It may help to use the fact that \(a^{-b}=\dfrac{1}{a^b}\).

A useful approximation is \(2^{10}=1,\!024\approx 1,\!000\).

Use the Fundamental Theorem of Calculus.

Calculate \(f'(x)\).

Using a Pythagorean identity makes this much easier.

Write the terms out. It should look very similar to a known series; make substitutions to make that series look like what you want.

Find the characteristic equation; it factors very nicely. Remember that each repeated root tacks on another factor of \(t\).

If \(f(x,y)\le g(x,y)\), then \(0\le g(x,y)-f(x,y)\). Use this fact to determine the integrand as well as the shape of the boundary.

You can calculate the integral directly and substitute the answers, but it may be easier to notice that \(f\) is constant on \([0,a]\), and one answer lends itself very nicely to this situation.

The order you multiply in matters. Multiply on either side (pick one) by however many copies of \(b\) or \(c\) you need to get the identity back.

Try sketching a few possible graphs. Remember, the graphs can get wild as long as they’re differentiable and go through \((0,1)\) and \((1,0)\).

Events \(A\) and \(B\) are independent if and only if \(\text{Pr}(A\cap B)=\text{Pr}(A)\text{Pr}(B)\).

The given condition is called Lipschitz continuity, and it essentially means the slope between any two points of the function is bounded. Does this imply differentiability?

Two vectors are orthogonal if their dot product is zero.

The solid of revolution should still contain the curve \(z=f(y)\) when \(x=0\).

Subspaces need to be closed under addition and scalar multiplication, and must contain the zero vector.

Split this into a difference of two summations.

Let \(n=3\) to make this problem more manageable to just work through.

Note that \(y=\sqrt{1-x^2}\) is the equation of the top half of the unit circle.

Subfields need to be closed under addition, subtraction, multiplication, and division.

Count the number of permutations of \(n\) apples. How many of these permutations have the apples lined up perfectly in order?

Use the Fundamental Theorem of Calculus. Here’s a powerful corollary worth memorizing:
\[\dfrac{d}{dx}\displaystyle\int_{a(x)}^{b(x)} f(t)\,dt=f\big(b(x)\big)b'(x)-f\big(a(x)\big)a'(x)\]

Try picking numbers for \(f(0)\) and \(f(2)\), and see what happens to \(f\) from there. Picking different examples can help you eliminate not-necessarily-true statements.

Start off with \(F(2,2)=F(1+1,1+1)\) and work from there.

Some facts you need to know about row operations and determinants:

1. Multiplying a row by a constant multiplies the determinant by that constant.
2. Switching two rows multiplies the determinant by \(-1\).
3. Adding a multiple of one row to another row does not change the determinant.

This is a Riemann sum. Identify the function being integrated and the interval over which it’s being integrated. (What is \(\dfrac{3i}{n}\) when \(i=1\) and \(n\) is large? What about when \(i=n\)?)

The logarithm of a negative number is nonreal. What values of \(x\) make this happen?

This region can be described as \(0\le y\le 1\), \(0\le z\le 1\), \(yz\le x\le 1\). Find the volume of this region using a double or triple integral.

Either calculate \(\begin{pmatrix}1&2\\0&-1\end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix}\) directly, or solve \((A-I)X=0\).

Create a second-order Maclaurin polynomial for \(f\) using the given data, and use the Lagrange error bound.

Rearrange the equation to one side, and use the Intermediate Value Theorem.

This is similar to the limit definition of the second derivative, but you can just hit this with L’Hôpital’s Rule twice.

Solve this as if it were a linear equation — how many solutions for \((x^2,y^3)\) are there? How many solutions does this make for \((x,y)\)?

There are two quick ways to do this. You could substitute values such as \(x=2\) and \(x=3\) and narrow down the choices from there, or you could think of it as a Taylor series, where \(a_n=\dfrac{\left.\frac{d^n}{dx^n}(x^3-x-1)\right|_{x=2}}{n!}\).

Use Green’s Theorem: \(\displaystyle\oint_{\partial\Omega} (P\,dx+Q\,dy)=\displaystyle\iint_\Omega (Q_x-P_y )\,dA\).

A product of two cyclic groups is cyclic if and only if the orders are coprime.

There’s a nice formula for variance: \(\text{Var}[X]=\text{E}[X^2]-(\text{E}[X])^2\).  Remember that \(\text{E}[X^n]=\int_{-\infty}^\infty x^n f(x)\,dx\).

Expanding by minors along the first row isn’t too hard, though you have to be careful. Alternatively, if the determinant is zero, then the first column could be expressed as a linear combination of the other three.

Note that \(\phi(0)=0\) and \(\phi(1)=1\). (Why?) Use this to extend \(\phi\) to the integers and then the rational numbers.

The first column of \(A\) is what \(A\) does to the vector \(\begin{pmatrix}1\\0\end{pmatrix}\), and the second column is what it does to \(\begin{pmatrix}0\\1\end{pmatrix}\).

Cauchy’s integral formula implies that \(\displaystyle\oint_C (z-a)^n\,dz=0\) for any counterclockwise-oriented curve \(C\) enclosing \(a\) and almost any integer \(n\). The only exception is that when \(n=-1\), the integral comes out to \(2\pi i\).

What can you say about \(f'(x)-g'(x)\)? What about \(f(x)-g(x)\)?

Any proper subgroup of the integers is of the form \(n\mathbb{Z}\), that is, the set of integer multiples of \(n\).

Good examples to try are the empty set, the rational numbers, and the irrational numbers.

Let \(n\) be a power of \(2\), like \(32\), and think about what happens. Alternatively, notice that this algorithm is faster than just scanning the whole list looking for the name.

You can calculate the characteristic equation: \(\det⁡(A-\lambda I)=0\), or you can just use the determinant and trace of \(A\).

Use Lagrange’s Theorem: If \(H\) is a subgroup of \(G\), then the order of \(H\) divides the order of \(G\).

Draw a Venn diagram. How many non-overlapping regions are there? (These are called the “atomic” sets.) How many ways can these be unioned together?

Think of any valid path as an arrangement of a word EEEEENNNNNNN — that is, consisting of five E’s and seven N’s.

Note that \([0,1]=[0,\frac{1}{2})\cup [\frac{1}{2},\frac{2}{3})\cup [\frac{2}{3},\frac{3}{4})\cup \ldots \cup \{1\}\). As for being Hausdorff or not, note that any open interval in the standard topology is still open in our new topology.

Rewrite the integral in polar coordinates. Remember that the Jacobian is \(r\).

If \(f\) is an eigenvector of \(D\), then \(Df=af\) for some constant \(a\). What functions have this property?

One of the words in the answer choices actually means “having a complex power series expansion”.

Use the Pigeonhole Principle. Try letting \(n=5\) to see which choices can be avoided.