Hints:  GR9367

Click on a question to uncover the hint.

Question 1

Substitute \(g(x)\) for \(x\) in the definition of \(f(x)\).

Question 2

DON’T use L’Hôpital’s Rule! The function is continuous at \(x=0\).

Question 3

Just calculate the integral directly. Watch out for the \(2\), and make sure to plug in \(0\) correctly.

Question 4

Draw a Venn Diagram for \(A\) and \(B\), and reason about how \(B\) must be related to \(A\). (Does one set have to live inside the other?)

Question 5

Yes, \(|x|\) is differentiable as long as \(x\neq 0\): its derivative is \(-1\) if \(x<0\) and \(1\) if \(x>0\).

Question 6

You could do a bunch of calculus, but all you need to do is sketch \(y=x^2+x\) (what are its roots?) and look at the graph.

Question 7

Remember that \(e^x>0\) for all real \(x\).

Question 8

The expression under the square root needs to be positive. Make sure your intervals don’t contain anywhere that \(\tan ⁡x\) blows up to infinity.

Question 9

Try a \(u\)-substitution with the denominator.

Question 10

Don’t bother calculating \(f(x)\) — just try all the solutions.

Question 11

Calculate \(\dfrac{\partial \phi}{\partial y}\) first.

Question 12

Use L’Hôpital’s Rule.

Question 13

It may help to use the fact that \(a^{-b}=\dfrac{1}{a^b}\).

Question 14

A useful approximation is \(2^{10}=1,\!024\approx 1,\!000\).

Question 15

Use the Fundamental Theorem of Calculus.

Question 16

Calculate \(f'(x)\).

Question 17

Using a Pythagorean identity makes this much easier.

Question 18

Write the terms out. It should look very similar to a known series; make substitutions to make that series look like what you want.

Question 19

Find the characteristic equation; it factors very nicely. Remember that each repeated root tacks on another factor of \(t\).

Question 20

If \(f(x,y)\le g(x,y)\), then \(0\le g(x,y)-f(x,y)\). Use this fact to determine the integrand as well as the shape of the boundary.

Question 21

You can calculate the integral directly and substitute the answers, but it may be easier to notice that \(f\) is constant on \([0,a]\), and one answer lends itself very nicely to this situation.

Question 22

The order you multiply in matters. Multiply on either side (pick one) by however many copies of \(b\) or \(c\) you need to get the identity back.

Question 23

Try sketching a few possible graphs. Remember, the graphs can get wild as long as they’re differentiable and go through \((0,1)\) and \((1,0)\).

Question 24

Events \(A\) and \(B\) are independent if and only if \(\text{Pr}(A\cap B)=\text{Pr}(A)\text{Pr}(B)\).

Question 25

The given condition is called Lipschitz continuity, and it essentially means the slope between any two points of the function is bounded. Does this imply differentiability?

Question 26

Two vectors are orthogonal if their dot product is zero.

Question 27

The solid of revolution should still contain the curve \(z=f(y)\) when \(x=0\).

Question 28

Subspaces need to be closed under addition and scalar multiplication, and must contain the zero vector.

Question 29

Split this into a difference of two summations.

Question 30

Let \(n=3\) to make this problem more manageable to just work through.

Question 31

Note that \(y=\sqrt{1-x^2}\) is the equation of the top half of the unit circle.

Question 32

Subfields need to be closed under addition, subtraction, multiplication, and division.

Question 33

Count the number of permutations of \(n\) apples. How many of these permutations have the apples lined up perfectly in order?

Question 34

Use the Fundamental Theorem of Calculus. Here’s a powerful corollary worth memorizing:
\[\dfrac{d}{dx}\displaystyle\int_{a(x)}^{b(x)} f(t)\,dt=f\big(b(x)\big)b'(x)-f\big(a(x)\big)a'(x)\]

Question 35

Try picking numbers for \(f(0)\) and \(f(2)\), and see what happens to \(f\) from there. Picking different examples can help you eliminate not-necessarily-true statements.

Question 36

Start off with \(F(2,2)=F(1+1,1+1)\) and work from there.

Question 37

Some facts you need to know about row operations and determinants:

  1. Multiplying a row by a constant multiplies the determinant by that constant.
  2. Switching two rows multiplies the determinant by \(-1\).
  3. Adding a multiple of one row to another row does not change the determinant.
Question 38

This is a Riemann sum. Identify the function being integrated and the interval over which it’s being integrated. (What is \(\dfrac{3i}{n}\) when \(i=1\) and \(n\) is large? What about when \(i=n\)?)

Question 39

The logarithm of a negative number is nonreal. What values of \(x\) make this happen?

Question 40

This region can be described as \(0\le y\le 1\), \(0\le z\le 1\), \(yz\le x\le 1\). Find the volume of this region using a double or triple integral.

Question 41

Either calculate \(\begin{pmatrix}1&2\\0&-1\end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix}\) directly, or solve \((A-I)X=0\).

Question 42

Create a second-order Maclaurin polynomial for \(f\) using the given data, and use the Lagrange error bound.

Question 43

Rearrange the equation to one side, and use the Intermediate Value Theorem.

Question 44

This is similar to the limit definition of the second derivative, but you can just hit this with L’Hôpital’s Rule twice.

Question 45

Solve this as if it were a linear equation — how many solutions for \((x^2,y^3)\) are there? How many solutions does this make for \((x,y)\)?

Question 46

There are two quick ways to do this. You could substitute values such as \(x=2\) and \(x=3\) and narrow down the choices from there, or you could think of it as a Taylor series, where \(a_n=\dfrac{\left.\frac{d^n}{dx^n}(x^3-x-1)\right|_{x=2}}{n!}\).

Question 47

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

Question 48

A product of two cyclic groups is cyclic if and only if the orders are coprime.

Question 49

There’s a nice formula for variance: \(\text{Var}[X]=\text{E}[X^2]-(\text{E}[X])^2\).  Remember that \(\text{E}[X^n]=\int_{-\infty}^\infty x^n f(x)\,dx\).

Question 50

Expanding by minors along the first row isn’t too hard, though you have to be careful. Alternatively, if the determinant is zero, then the first column could be expressed as a linear combination of the other three.

Question 51

Note that \(\phi(0)=0\) and \(\phi(1)=1\). (Why?) Use this to extend \(\phi\) to the integers and then the rational numbers.

Question 52

The first column of \(A\) is what \(A\) does to the vector \(\begin{pmatrix}1\\0\end{pmatrix}\), and the second column is what it does to \(\begin{pmatrix}0\\1\end{pmatrix}\).

Question 53

Cauchy’s integral formula implies that \(\displaystyle\oint_C (z-a)^n\,dz=0\) for any counterclockwise-oriented curve \(C\) enclosing \(a\) and almost any integer \(n\). The only exception is that when \(n=-1\), the integral comes out to \(2\pi i\).

Question 54

What can you say about \(f'(x)-g'(x)\)? What about \(f(x)-g(x)\)?

Question 55

Any proper subgroup of the integers is of the form \(n\mathbb{Z}\), that is, the set of integer multiples of \(n\).

Question 56

Good examples to try are the empty set, the rational numbers, and the irrational numbers.

Question 57

Let \(n\) be a power of \(2\), like \(32\), and think about what happens. Alternatively, notice that this algorithm is faster than just scanning the whole list looking for the name.

Question 58

You can calculate the characteristic equation: \(\det⁡(A-\lambda I)=0\), or you can just use the determinant and trace of \(A\).

Question 59

Use Lagrange’s Theorem: If \(H\) is a subgroup of \(G\), then the order of \(H\) divides the order of \(G\).

Question 60

Draw a Venn diagram. How many non-overlapping regions are there? (These are called the “atomic” sets.) How many ways can these be unioned together?

Question 61

Think of any valid path as an arrangement of a word EEEEENNNNNNN — that is, consisting of five E’s and seven N’s.

Question 62

Note that \([0,1]=[0,\frac{1}{2})\cup [\frac{1}{2},\frac{1}{3})\cup [\frac{1}{3},\frac{1}{4})\cup \ldots \cup \{1\}\). As for being Hausdorff or not, note that any open interval in the standard topology is still open in our new topology.

Question 63

Rewrite the integral in polar coordinates. Remember that the Jacobian is \(r\).

Question 64

If \(f\) is an eigenvector of \(D\), then \(Df=af\) for some constant \(a\). What functions have this property?

Question 65

One of the words in the answer choices actually means “having a complex power series expansion”.

Question 66

Use the Pigeonhole Principle. Try letting \(n=5\) to see which choices can be avoided.

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