The equation of the tangent line is just the first-order Taylor polynomial.

Think about how the dimension of \(V\cap W\) affects the number of distinct basis vectors of \(V\) and \(W\). It may help to think of the situation in \(\mathbb{R}^3\) first, and then consider what happens in \(\mathbb{R}^4\).

Use the Intermediate Value Theorem, or write the equation as \(e^x=2-x\) and sketch a graph.

The function must be of the form \(f(x)=a(x-2)^2\).

Sketch the graphs as accurately as you can. Remember that the equation of a circle is \((x-h)^2+(y-k)^2=r^2\).

Don’t bother actually integrating this; instead sketch the graph and use geometry.

Use the triangle area formula \(\text{Area}=\dfrac{1}{2}ab\sin ⁡C\).

Arrange the functions themselves in order, and the rest follows from integration properties. Remember that \(0\le x\le 1\).

The function \(g\) must be an antiderivative of \(g’\), so use area to solve the problem.

Use the approximation \(1.5(266)=399\approx 400\).

Suppose the first row of the matrix is \(\begin{pmatrix}a&b\end{pmatrix}\). Complete the rest of the matrix and try the eigenvector candidates.

Draw the rectangle so that \(w+l+w=x\) and use derivatives. Be careful … \(x\) is a constant for this problem!

Calculate successive powers of \(7\), modding out by \(10\) whenever possible, to find a pattern.

Think about the relationship between continuity, integrability, and differentiability.

Use the Disk Method: \(V=\pi\displaystyle\int_a^b R(x)^2\,dx\).

Take the derivative. What does this tell you about the shape of the graph?

Use the Rank-Nullity Theorem: \(\dim ⁡V=\text{rank }T + \text{nullity }T\).

Try integrating both sides, or moving everything to one side of the equation.

Sketch the graph as two “blurry” lines. Where if anywhere could the resulting graph possibly be continuous?

For each \(P_m\cap P_n\), compare the prime factorizations of \(m\) and \(n\).

Subspaces must be closed under addition and scalar multiplication, and must contain the zero vector (function).

For the graphs to be tangent, the functions must be equal and so must their derivatives. Use some substitution to figure it out from there.

Split the integrand into two terms so that one can be factored out of the integral. Then use the Fundamental Theorem of Calculus.

Write out the first few terms and look for a pattern.

Set the gradient equal to zero, and use the Hessian to classify the extrema. (Technically, though, you don’t have to — there are three answers that imply one of the others, and therefore can’t possibly be correct based on other information you know!)

Think about what the dimension of the intersection must be. Then just try a value.

Just start deleting edges and make sure you don’t disconnect the graph.

Plug each set of functions into the definition. One of them can’t be reduced to \(f\sim g\) but gives something else instead.

Use De Morgan’s Laws.

Try substituting various values of \(y\).

The fact that \(\oplus\) is commutative and \(\odot\) isn’t matters a lot.

Work your way through the algorithm. Once you’ve got the first two numbers, you’re done.

Sketch two spheres, and draw a line segment connecting their centers.

How many ways can we arrange the men? How many way can we arrange the guests if we then glue the men together in order?

All but one answer choice is equivalent to \(M\) being invertible. The wrong answer is very subtle and isn’t quite strong enough.

Let \(z=a+bi\) and solve the equation. Also notice that \(|z|\) must be real.

Try letting \(f(x)=x^2\) and letting \(C\) and \(D\) be particular closed intervals.

As you drag the rightmost point to infinity, the sides of length \(r\) and \(s\) become more parallel, but it seems that one is always longer than the other. How much longer?

All fields are integral domains. For C, don’t forget about piecewise functions.

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

Find the probability of the complementary event: both \(x\) and \(y\) are less than or equal to \(3\).

What do the \(n\)th roots of unity always add up to?

Use the normal approximation to the binomial distribution: \(\mu=np\) and \(\sigma =\sqrt{np(1-p)}\). Also remember the Empirical Rule (\(68\%\), \(95\%\), \(99.7\%\)).

Use the Pigeonhole Principle.

Since \(G\) is cyclic, any homomorphism \(\varphi: G\to G\) is entirely determined by what \(\varphi\) does to a generator of \(G\).

Use the formula \(\text{Work}=\displaystyle\int _C \mathbf{F}\cdot d\mathbf{r}=\displaystyle\int _a^b\mathbf{F}(\mathbf{r}(t)) \mathbf{r}'(t)\,dt\)\.

The way the problem is phrased actually implies that \(f\) is continuous and differentiable without outright saying it.

Use the Fundamental Theorem of Finite Abelian Groups.

Try matrices with small positive and negative numbers. Also remember that \(\det⁡(AB)=\det⁡(A) \det⁡(B)\).

This problem is a huge time sink. Write this as a sum of integrals over unit intervals. To find a formula for the power series you need, start with a geometric series, and try shifting it by multiplying or taking derivatives.

(If you know a faster way to solve this problem, contact me!)

To eliminate answer choices, try to find sets that contain all rational numbers but don’t contain all real numbers.

Use Lagrange Multipliers. Alternatively, use spherical symmetry and remember that the gradient of a function is the direction of greatest increase.

Draw radii and look for isosceles right triangles. Also, try coming up with “Figure 0” to make the question even easier.

The number of zeros at the end of \(n!\) is \(\left\lfloor\dfrac{n}{5}\right\rfloor +\left\lfloor\dfrac{n}{25}\right\rfloor +\left\lfloor\dfrac{n}{125}\right\rfloor +\cdots\). Note that this looks remarkably like a geometric series; use this to approximate one value, and nudge from there.

One of the formulas breaks the Triangle Inequality with very simple numbers.

Use the Ratio Test. Alternatively, use Stirling’s approximation \(n!\sim\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n\) along with \(\lim\limits_{n\to\infty}\sqrt[n]{n}=1\).

Use the definition of similar matrices: \(A\) is similar to \(B\) if there exists \(P\) such that \(B=PAP^{-1}\).

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

Look for lines of reflectional symmetry and angles of rotational symmetry.

Remember that \(\mathbb{R}\) has cardinality \(\mathfrak{c}=2^{\aleph_0}\). Also, remember that finite unions of infinite sets of a given cardinality have the same cardinality, and that the power set of a set always has a strictly larger cardinality.

Use the Heine-Borel Theorem: In \(\mathbb{R}^n\), a set is compact if and only if it is closed and bounded.

Take the derivative, but don’t forget to use the limit definition of the derivative for \(x=0\).

Uniform convergence preserves continuity. For III, use Lebesgue’s Dominated Convergence Theorem.

Try sketching functions out. Remember the Heine-Borel Theorem: In \(\mathbb{R}^n\), a set is compact if and only if it is closed and bounded.

Think of quaternions and modular arithmetic.