This is an arithmetic sequence. Look for a pattern.

The inverse of \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is \(A^{-1}=\dfrac{1}{\det ⁡A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\).

Solve the equation for \(b\) first using antiderivatives. Look for where \(y=x\) and \(y=x^2\) intersect.

The graph of \(f’\) needs to describe the slope of \(f\) (i.e. increasing/decreasing behavior and relative extrema).

Follow your way through the instructions. It may help to replace \(999\) with a small number like \(10\) and reason accordingly.

Check the points at \(t=-\dfrac{\pi}{2}\) and \(t=0\).

Make a \(u\)-substitution using the denominator.

A one-to-one function from a set to itself is also called a permutation.

Sketch the two branches as “blurry” curves. Where could the function possibly be continuous?

Substitute (distinct) values of \(x\) and \(y\).

Come up with an interval that satisfies the condition.

Consider the three different cases and add them together.

When negating quantifiers, \(\exists\) and \(\forall\) switch: \(\neg \Big(\exists x\ \forall y\ P(x,y)\Big)\) is equivalent to \(\forall x\ \exists y\ \neg P(x,y)\).

Sketch \(y=a\) and \(y=x\), and try different graphs. (It may help to let \((a,b)=(-1,1)\).)

Write out the linear combination explicitly and solve for the coefficients.

Construct the table, working your way downward and to the right.

Draw and label the radii. The area of an annulus is just the difference of the areas of two circles.

In a group, every element must have an inverse, and that inverse must be unique.

Use the limit definition of the derivative, and try various functions.

One way to make this go quicker is to remember that the tangent line is just the first-order Taylor polynomial.

Remember that subgroups must be closed under the operation and contain inverses.

Sketch the situation, and make sure to label the points so you’re giving the correct angle.

Two nonzero vectors are orthogonal if their dot product is zero.

Since \(f\) is linear, the maximum must occur on one of the vertices of the feasible region.

A local maximum can only occur when \(f’\) is zero or undefined. Be careful that you check the domain of each branch.

Use the Chain Rule.

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

Use integration by parts.

Any basis for \(\mathbb{R}^n\) must consist of exactly \(n\) nonzero linearly independent basis vectors.

Use the Rational Roots Theorem.

What’s the relationship between the number of arrangements that have Pat ahead of Lynn and the number that don’t?

Finding \(\text{lcm}(30,16)\) will come in handy.

Think of a function with a horizontal asymptote of \(y=1\).

The tangent plane to \(z=f(x,y)\) has normal vector \((f_x,f_y,-1)\). Alternatively, if you solve the answers for \(z\), the gradient of the surface has to equal the gradient of the tangent plane.

Put the first four numbers in order and try different values of \(x\).

Cancel out one thing immediately, and notice that \(k=(k-1)+1\).

Sketching the graph of each function will go much more quickly than actually integrating.

Answer choices A-C are different kinds of rectangular sums. Note that \(e^{-x}\) is decreasing.

Notice that \(\text{Pr}(H>T)+\text{Pr}(H=T)+\text{Pr}(H<T)=1\).

For \(f\) to have a relative maximum, you need \(\nabla f=0\), \(\det H_f>0\) (where \(H_f=\begin{pmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{pmatrix}\) is the Hessian matrix), and \(f_xx<0\).

Sketch the points and think of translating the vectors between those points.

Represent \(\begin{pmatrix}6\\7\\8\end{pmatrix}\) as a linear combination of \(\begin{pmatrix}0\\1\\2\end{pmatrix}\) and \(\begin{pmatrix}3\\4\\5\end{pmatrix}\). (The \(0\) helps.) Then use linearity.

Use logarithmic differentiation.

Check the units to make sure that you get gallons. (Remember that \(dt\) has a unit too!)

You could find the characteristic equation, but the trace makes this so much easier.

Sketch the region \([0,1]\times [0,1]\), and shade the region where \(|y-x|<\dfrac{1}{2}\).

Look at the potential ranges of \(u\) and \(v\). Then figure out where the boundaries of \([0,1]\times [0,1]\) map to under the change of variables.

You can either use substitutions and integration rules, or you can think of these geometrically (i.e. transformations).

Note that if \(y^2=x^2\), then \(|y|=|x|\). (Why?) Think about how many ways the branches can be arranged.

Use the Ratio Test.

Use row reduction. Note that the system is homogeneous, so think of easy solutions.

Sketch the circle, and use residue integration: \(\displaystyle\oint_C f(z)\,dz=2\pi i\cdot \displaystyle\sum_{i=1}^n \text{Res}(f,z_i)\) , where each \(z_i\) is a pole of \(f\) enclosed within \(C\).

Set up the differential equation, and note that at in the limit, \(h'(t)=0\).

Sketch a tiny bit of the curve at \(x=0\), paying attention to slope and concavity, and then use the fact that \(f^{\prime\prime}\) is increasing without bound to sketch the rest of \(f\).

A metric must be positive definite (i.e. nonnegative, and zero only when two points are the same), symmetric, and satisfy the triangle inequality. To test the triangle inequality, try some simple points.

Subrings must be closed under addition, subtraction, and multiplication.

Try sketching a few functions over \((0,1)\), and remember that continuous functions preserve connectedness and compactness.

Exactly two of the elements in the set must be equal. Think about what must equal the identity, and keep Lagrange’s Theorem in mind: the order of an element must divide the order of a group.

Try replacing \(s\) with \(s+s\) or \(s+t\) and playing with the algebra.

Factor the expression, and remember that \(p\) must be an odd prime. Also use Fermat’s Little Theorem: If \(p\) does not divide \(a\), then \(a^{p-1}\equiv 1\pmod{⁡p}\).

Just focus on the terms that will multiply to give you \(x^3\): \(x^3\cdot x^0\) and \(x^1\cdot x^2\). Use the Binomial Theorem to get the coefficients.

Sketch the graphs, but remember that exponential functions always eventually overtake polynomial functions.

Continuous functions on compact sets are uniformly continuous. However, Lipschitz continuity is stronger than uniform continuity.

Write \(p\) in the form \((x-r_1)(x-r_2)(x-r_3)\) and reason from there with the given information about \(p\) and \(p’\).

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