Hints:  GR0568

Click on a question to uncover the hint.

Question 1

This is just the parametrization of (some part of) a circle.

Question 2

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

Question 3

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 \(\mathbb{R}^4\).

Question 4

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

Question 5

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

Question 6

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

Question 7

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

Question 8

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

Question 9

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

Question 10

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

Question 11

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

Question 12

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.

Question 13

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

Question 14

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

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

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

Question 21

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

Question 22

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

Question 23

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.

Question 24

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

Question 25

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

Question 26

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!)

Question 27

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

Question 28

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

Question 29

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

Question 30

Use De Morgan’s Laws.

Question 31

Try substituting various values of \(y\).

Question 32

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

Question 33

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

Question 34

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

Question 35

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?

Question 36

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

Question 37

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

Question 38

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

Question 39

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?

Question 40

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

Question 41

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

Question 42

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

Question 43

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

Question 44

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

Question 45

Use the Pigeonhole Principle.

Question 46

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

Question 47

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

Question 48

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

Question 49

Use the Fundamental Theorem of Finite Abelian Groups.

Question 50

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

Question 51

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!)

Question 52

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

Question 53

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

Question 54

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

Question 55

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.

Question 56

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

Question 57

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

Question 58

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

Question 59

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

Question 60

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

Question 61

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.

Question 62

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

Question 63

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

Question 64

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

Question 65

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.

Question 66

Think of quaternions and modular arithmetic.

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