Hints:  GR1768

(Note:  This form is identical to form GR1268.  The only difference is that the directions have been updated to reflect the fact that points are no longer deducted for incorrect answers.)

Click on a question to uncover the hint.

Question 1

Use L’Hôpital’s Rule.

Question 2

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

Question 3

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

Question 4

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

Question 5

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

Question 6

Raise all three numbers to the same power.

Question 7

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

Question 8

One of the choices isn’t closed under inversion.

Question 9

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

Question 10

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

Question 11

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

Question 12

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

Question 13

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

Question 14

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

Question 15

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

Question 16

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

Question 17

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 \((0,1)\).

Question 18

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

Question 19

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

Question 20

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

Question 21

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

Question 22

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.

Question 23

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

Question 24

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

Question 25

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

Question 26

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

Question 27

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

Question 28

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

Question 29

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.

Question 30

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

Question 31

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.

Question 32

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

Question 33

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

Question 34

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

Question 35

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

Question 36

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

Question 37

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

Question 38

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

Question 39

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.

Question 40

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

Question 41

Find the cross product of the normals to the plane.

Question 42

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

Question 43

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

Question 44

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

Question 45

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

Question 46

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

Question 47

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

Question 48

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?

Question 49

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

Question 50

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.

Question 51

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.

Question 52

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

Question 53

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.

Question 54

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

Question 55

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!)

Question 56

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.

Question 57

Uniformly continuous functions preserve Cauchy sequences.

Question 58

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.

Question 59

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.

Question 60

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.

Question 61

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.

Question 62

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.

Question 63

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

Question 64

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: \({\large\bigcirc}\kern-1.55em\displaystyle\iint_{\partial T} \mathbf{F}\cdot\hat{\mathbf{n}}\,dS = \displaystyle\iiint_T \nabla\cdot\mathbf{F}\,dV\).

Question 65

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 66

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.

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