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# Hints:  GR0568

Click on a question to uncover the hint.

Solutions can be found here:

##### 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.

5 + 3 =