# Hints: GR8767

Click on a question to uncover the hint.

Solutions can be found here:

##### Question 1

Form vectors by connecting \((0,0,0)\) to the other two points; the plane is spanned by these two vectors.

##### Question 2

Be sure to consider cases in which one or both of \(a\) and \(b\) are negative.

##### Question 3

Calculate the inside integral first, treating \(x\) as a constant, then calculate the outside integral.

##### Question 4

Use the Product Rule and combine like terms.

##### Question 5

You could reconstruct the function from its gradient, but it’s much easier to just look at the partial derivatives of the answers and use process of elimination.

##### Question 6

The derivative along each linear piece is constant. Pay attention to the signs of those slopes.

##### Question 7

Look for two functions that bound the shaded region above and below.

##### Question 8

What happens to the terms as \(n\) approaches infinity?

##### Question 9

First find the probability that \(0\) is not chosen for one of the digits, then figure out how repeating this process \(k\) times affects your answer.

##### Question 10

The agent has been given \(MC\); what do you need to multiply this by to get \(M\) by itself, and on which side?

##### Question 11

Solve for \(x\) in the first equation and calculate its value. Then find an angle in the range \([0,\pi]\) whose cosine is that value.

##### Question 12

Rewrite the integrand as a single power of \(e\).

##### Question 13

Remember the formula to factor a sum of cubes: \(a^3+b^3=(a+b)(a^2-ab+b^2)\).

##### Question 14

How should the total percent change for city A’s electricity use compare to the percent changes in household use and in other uses? What is the range of possible values for the total percent change?

##### Question 15

Represent \((3,5)\) as a linear combination of \((1,1)\) and \((-1,0)\). (The zero helps with this.) Then use the linearity of \(f\).

##### Question 16

Sketch the situation, and think of secant lines and tangent lines. The answers describe various theorems you learned in calculus (Intermediate Value Theorem, Mean Value Theorem, etc.).

##### Question 17

Check the properties directly. Does swapping \(a\) and \(b\) change the value? Does \(a+e+2ae=a\) have a solution for \(e\)? If so, does \(a+b+2ab=e\) have a solution for \(b\)?

##### Question 18

Write \((ab)^2\) as \(abab\) and see what you can do to the equation.

##### Question 19

Look for where \(f'(x)=0\), and be careful to check the neighboring values to make sure it’s a minimum.

##### Question 20

If a polynomial is periodic, it must be a very special kind of polynomial.

##### Question 21

What can you substitute for \(x\) in order to get \(f(x)\) on the left-hand side?

##### Question 22

If you happen to know the antiderivative of \(\dfrac{1}{\sqrt{1-x^2}}\) (it’s an inverse trig function), you can work this pretty quickly. Otherwise you can substitute \(x=\sin u\).

##### Question 23

Think of the contrapositive of \(S(k)\to S(k+1)\).

##### Question 24

Work your way through the definition: plug \(1\) into \(g\), figure out which branch of the function that requires you to go to, and go from there.

##### Question 25

Every multiple of \(3x+7y\) will also be divisible by \(11\). Be on the lookout for alternate ways to represent coefficients mod \(11\) (for example, \(10\equiv -1\)).

##### Question 26

This is a variant of the topologist’s sine curve. Sketch the graph, and think of the different placements of \((0,k)\).

##### Question 27

For \(f\) to be differentiable, both branches of \(f\) and \(f’\) must match at \(x=1\).

##### Question 28

Remember that an integral is just an area between a curve and the \(x\)-axis. Just think of geometry.

##### Question 29

Sketch the graph, and think about what will happen to the points where \(x=a\), \(x=b\), and \(x=\dfrac{a+b}{2}\).

##### Question 30

Horizontal translation doesn’t affect the value of the integral, so let \([a,b]=[-r,r]\) and notice that \(f\) is then even. (What does that mean about \(f’\)?)

##### Question 31

Use L’Hôpital’s Rule.

##### Question 32

Eliminate any vectors that can be written as linear combinations of the other vectors. The zero vector is redundant.

##### Question 33

Note that \((0,a)\) is in the shaded region for any \(a\ge 0\).

##### Question 34

Sketch the situation and look for similar triangles.

##### Question 35

Subtract row 1 from row 2. Try adding multiples of this result to row 1.

##### Question 36

You could set up a bunch of calculus, but notice that the curve \(xy=a\) (or \(y=\dfrac{a}{x}\)) is a hyperbola with symmetry about the lines \(y=x\) and \(y=-x\).

##### Question 37

The valid steps can be arrived at from one another by some kind of substitution or other algebraic operation. For example, (1) implies (2) because we can replace \(x\) with \(-x\) on both sides. Look for an implication — forwards or backwards — that can’t be justified.

##### Question 38

Let \(\mathbf{x}=\begin{pmatrix}a\\b\\c\end{pmatrix}\), and calculate \(M\mathbf{x}\). What would happen if you applied \(M\) a total of \(100\) times?

##### Question 39

Think about what \(\dfrac{|x|}{x}\) equals when \(x>0\) and \(x<0\) to sketch the graph.

##### Question 40

This is a first-order linear equation, so you could use an integrating factor. However, notice the expression \(x\,dy+y\,dx\); can you do something with that to make it easier?

##### Question 41

Calculate \(y'(1)\) and \(y^{\prime\prime}(1)\).

##### Question 42

Remember that \(\mathbf{u}\cdot\mathbf{v}=\Vert\mathbf{u}\Vert\Vert\mathbf{v}\Vert\cos{\theta}\).

##### Question 43

If a polynomial has real coefficients, then any complex roots must come in conjugate pairs.

##### Question 44

You could cleverly add and subtract the same term from the numerator, but it’s much faster to just use L’Hôpital’s Rule.

##### Question 45

You can use the Ratio Test, but you can also collect the \(e^n\) and \(x^n\) into a single term and recognize a known Maclaurin series with a known radius of convergence.

##### Question 46

Take the logarithm of both sides.

##### Question 47

Use the Pigeonhole Principle, and remember that a set of size \(n\) has \(2^n\) possible subsets, including itself and the empty set.

##### Question 48

Calculate \(S(T(p(x)))\) and \(T(S(p(x)))\). (Put in plain English, \(S\) takes the derivative of a polynomial, and \(T\) multiplies it by \(x\).)

##### Question 49

Use Lagrange’s Theorem: the order of a group element (or equivalently the order of the subgroup it generates) must divide the order of the whole group.

##### Question 50

Note that the possible winning sequences for player 1 are \(H\), \(TTH\), \(TTTTH\), and so on. Find the sum of all these probabilities.

##### Question 51

If \(L=\) “\(x_\infty\)” is the limit, then the limit should also satisfy the recursive equation.

##### Question 52

You can use the determinant and the trace to find the eigenvalues quickly. Also, note that the sum of the rows is the same, so \(\begin{pmatrix}1\\1\end{pmatrix}\) is an eigenvector.

##### Question 53

If \(p\in W\), then \(p\) must be a scalar multiple of \(x(x-1)(x+1)\). Remember that \(n\)th degree polynomials have \(n+1\) terms.

##### Question 54

If this map (call it \(\varphi\)) is a group homomorphism, then \(\varphi(xy)=\varphi(x)\varphi(y)\).

##### Question 55

Critical points occur where \(\nabla f=0\). The Hessian \(H_f=\begin{pmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{pmatrix}\) can classify them: If \(\det{H_f}<0\), you have a saddle point, and if \(\det{H_f}>0\), you have a minimum if \(f_{xx}>0\) and a maximum if \(f_{xx}<0\) (just like in the two-dimensional case for \(f^{\prime\prime}\)).

##### Question 56

Use the next term in the Taylor series for \(\sqrt{x}\) centered at \(x=1\).

##### Question 57

Figure out how many valid sequences there are for \(k=1\) and \(k=2\).

##### Question 58

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 59

Try subdividing the interval \([0,12]\) where anything “interesting” happens and calculating your sum.

##### Question 60

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 61

If \(A=A^{-1}\), what are the possible eigenvalues of \(A\)? (Think of the determinant.)

##### Question 62

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

##### Question 63

Sketch \(f^{-1}\) as a reflection of \(f\) over the line \(y=x\). You can answer this one without a single calculation — just compare areas.

##### Question 64

Think about what conditions are preserved by continuous functions. This is essential to know for the GRE.

##### Question 65

Factor the left-hand side like a quadratic and you’ll get something much nicer.

##### Question 66

Check for closure under addition, subtraction and multiplication.