Sketch it out, and use \(30^{\circ}:60^{\circ}:90^{\circ}\) triangles.

Make a \(u\)-substitution, and be careful with your limits of integration.

Think about how the dimension of \(V\cap W\) affects the number of distinct basis vectors of \(V\) and \(W\).

Write out the inadmissible outcomes. (There aren’t many.)

Raise all three numbers to the same power.

Use the sign of \(f’\) to determine when \(f\) is increasing or decreasing, and use the relative areas to rank the values.

One of the choices isn’t closed under inversion.

Look for where \(g\) must have a horizontal tangent line and where \(g\) must be concave up or concave down.

Think of those two expressions as distances from points, and sketch the solution.

Use the Washer Method: \(V=\pi\displaystyle\int_a^b \Big(R_{\text{outer}}(x)^2-R_{\text{inner}}(x)^2\Big)\,dx\).

Every group with prime order is cyclic. Otherwise use the Fundamental Theorem of Finite Abelian Groups.

Consider the limiting case where \(f'(x)=-1\) for all \(x\).

Use the Fundamental Theorem of Calculus, or substitute a particular value of \(x\) that will make the problem easy.

Try a combination of the functions \(x^2\) and \(e^x\).

Draw out a truth table for \(A\), \(B\), and \(C\). Be careful — there’s more than one way that \(C\) can be true.

A: Move everything to one side, and use calculus to determine the monotonic behavior.

B: Move everything to one side, and find the discriminant.

C: The lines have different slopes.

D,E: Sketch the graphs as accurately as possible. It may help to know that the graph of \(y=e^{-x^2}\) is a bell curve whose maximum is at \((0,1)\).

Differentiate the power series term by term — you should recognize the resulting power series.

Write everything in polar form \(re^{i\theta}\).

Use L’Hôpital’s Rule, or realize that the limit is a derivative in disguise.

Notice the symmetric interval, and think about even and odd functions.

You could calculate this integral directly, but you can make it easier with centroids, and make it even easier if you visualize it as a “lipstick-shaped” solid, slice off the top and rotate it to make a cylinder.

Notice that \((S,+,\cdot)\) is isomorphic to \(\mathbb{Z}_5\), which is a field.

Check D first, then reason about the others. Note that choice E implies that the corresponding matrix has rank \(1\).

An inflection point of \(h\) occurs where \(h^{\prime\prime}\) changes sign. What does this mean about the behavior of \(h’\)?

Notice that \(12\equiv 1\pmod{11}\).

You could use deMoivre’s Theorem, or you could calculate \((1+i)^2\) and go from there.

You have to just do the calculations — no way around it.

Just draw them all out. It may help to come up with all trees with \(4\) vertices first, and then base the trees with \(5\) vertices off those.

Sketch the graphs. If there is only one solution, the curves have to be tangent.

Notice all the rows add up to the same value, so that value has to be an eigenvalue. Then, use the trace, and inspect \(A-\lambda I\) for each remaining eigenvalue.

Use the Fundamental Theorem of Calculus, and factor out the “smaller” power of \(e\).

Write out the first few derivatives, and look for a pattern for the \(n\)th derivative.

In an upper triangular matrix, the values along the main diagonal are the eigenvalues. Use this fact to eliminate all the true statements.

The vector between the origin and the closest point on the plane must be normal to the plane.

What can you say about the interior of a set, in terms of being open, closed, connected, etc.?

Try small \(2\times 2\) matrices with \(1\)’s and \(0\)’s. Otherwise think of characteristic and minimal polynomials.

The sum of the exterior angles in a convex polygon is always \(360^{\circ}\).

Just run through the algorithm and be sure to keep track of the values of the variables. All you really need is the first two values printed by the algorithm.

Try combining example functions such as \(x^2\), \(2x\), and \(x+1\).

Find the cross product of the normals to the plane.

The metric given is the discrete metric, in which all sets are open.

Use the formulas \(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\) and \(\dfrac{d^2y}{dx^2}=\dfrac{dy’/dt}{dx/dt}\).

Find the equilibrium solutions by setting \(y’=0\) and reasoning using the signs of \(x\) and \(y\).

Show that the period of \(\cos⁡(97x)\) is approximately \(\dfrac{1}{15}\) and reason from there.

Sketch the situation, use the Pythagorean Theorem, and calculate the related rates.

Sketch the two branches of the function as “blurry” parabolas.

If \(\mathbf{u}\) is a unit vector, then the directional derivative in the direction of \(\mathbf{u}\) is \(\nabla g\cdot \mathbf{u}\). Then, between what two integers does \(|\mathbf{i}+2\mathbf{j}+3\mathbf{k}|\) lie?

The order of a product of an \(m\)-cycle and an \(n\)-cycle (assuming they are disjoint) is \(\text{lcm}(m,n)\).

Consider the ideal \((x,y)\) in \(\mathbb{R}[x,y]\) (the ring of real polynomials in \(x\) and \(y\)). For future reference, it’s worth memorizing how to combine ideals to get new ideals.

Eliminate the answers whose vectors aren’t orthonormal (i.e. unit length with pairwise zero dot product), then do just enough row reduction to find out which columns are pivot columns.

Assign the letters ABCDEFGHIJ to the professors, and think of the problem as an arrangement of the “word” AABBCCDDEEFFGGHHIIJJ.

Distribute in the integrand so that you can split it into two integrals (one of which can have \(x\) pulled out of it). Then use the Fundamental Theorem of Calculus to differentiate as many times as possible.

Draw the rectangle \([0,3]\times [0,4]\), and shade the region in which \(x<y\).

Long way: Note that the numerator equals \((1+e^{ax})-(1+e^{bx})\); use partial fractions to split into two integrals. Then for one of those integrals, substitute \(u=e^{bx}\) and use partial fractions again.

Short way: Let \(a=1\) and \(b=0\) and watch what happens. (Ignore that they said \(a\) and \(b\) should be positive — this is a limiting case!)

Solve for the constant in each inequality and take limits. For II, note that \(\displaystyle\sum_{k=1}^n k^2 \approx \dfrac{1}{3}n^3\), and for III, think of power series.

Uniformly continuous functions preserve Cauchy sequences.

Use the arc length formula \(s=\displaystyle\int _a^b \sqrt{\left(\dfrac{dx}{d\theta}\right)^2+\left(\dfrac{dy}{d\theta}\right)^2+\left(\dfrac{dz}{d\theta}\right)^2}\,d\theta\) and watch out for Pythagorean identities.

Use process of elimination. Some useful facts are that \(\det⁡(AB)=\det⁡(A)\det⁡(B)\) and that an invertible matrix must have full rank. Keep in mind, with these kinds of questions, the wrong answer can be very subtle, for instance mixing up “exists” statements and “all” statements.

In plain English, the property states that \(f(x)\) gets and stays further from \(f(1)\) as \(x\) gets further from \(1\).  Try to sketch such a function.

Set up your differential equation: \(\text{rate of change} = (\text{flow in}) – (\text{flow out})\). Also note that the solution will approach equilibrium, but never go past it; this can eliminate some answers.

Use the fact that the rationals and irrationals are both dense in the reals. Also consider the relationship between being connected and being path-connected.

Let \(A\) and \(B\) be various closed intervals to narrow down the choices. Make sure to consider negative numbers.

Calculate the flux directly using the formula \(\text{Flux}={\large\bigcirc}\kern-1.55em\displaystyle\iint_S \mathbf{F}\cdot\hat{\mathbf{n}}\,dS\), or use the Divergence Theorem: \({\large\bigcirc}\kern-1.55em\displaystyle\iint_{\partial T} \mathbf{F}\cdot\hat{\mathbf{n}}\,dS = \displaystyle\iiint_T \nabla\cdot\mathbf{F}\,dV\).

Use the Cauchy-Riemann equations: If \(f(z)=u(x,y)+iv(x,y)\) is analytic, then \(u_x=v_y\) and \(u_y=-v_x\).

Start calculating powers of \(5\), reducing mod \(17\) whenever possible. Also, note that \(\mathbb{Z}_{17}^{\times}\) has order \(16\), so Lagrange’s Theorem limits the possible orders of elements.