Hints:  GR9768

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Question 1

Just calculate the integral directly. No tricks here.

Question 2

This is an arithmetic sequence. Look for a pattern.

Question 3

The inverse of \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is \(A^{-1}=\dfrac{1}{\det ⁡A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\).

Question 4

Solve the equation for \(b\) first using antiderivatives. Look for where \(y=x\) and \(y=x^2\) intersect.

Question 5

The graph of \(f’\) needs to describe the slope of \(f\) (i.e. increasing/decreasing behavior and relative extrema).

Question 6

Follow your way through the instructions. It may help to replace \(999\) with a small number like \(10\) and reason accordingly.

Question 7

Check the points at \(t=-\dfrac{\pi}{2}\) and \(t=0\).

Question 8

Make a \(u\)-substitution using the denominator.

Question 9

A one-to-one function from a set to itself is also called a permutation.

Question 10

Sketch the two branches as “blurry” curves. Where could the function possibly be continuous?

Question 11

Substitute (distinct) values of \(x\) and \(y\).

Question 12

Come up with an interval that satisfies the condition.

Question 13

Consider the three different cases and add them together.

Question 14

When negating quantifiers, \(\exists\) and \(\forall\) switch: \(\neg \Big(\exists x\ \forall y\ P(x,y)\Big)\) is equivalent to \(\forall x\ \exists y\ \neg P(x,y)\).

Question 15

Sketch \(y=a\) and \(y=x\), and try different graphs. (It may help to let \((a,b)=(-1,1)\).)

Question 16

Write out the linear combination explicitly and solve for the coefficients.

Question 17

Construct the table, working your way downward and to the right.

Question 18

Draw and label the radii. The area of an annulus is just the difference of the areas of two circles.

Question 19

In a group, every element must have an inverse, and that inverse must be unique.

Question 20

Use the limit definition of the derivative, and try various functions.

Question 21

One way to make this go quicker is to remember that the tangent line is just the first-order Taylor polynomial.

Question 22

Remember that subgroups must be closed under the operation and contain inverses.

Question 23

Sketch the situation, and make sure to label the points so you’re giving the correct angle.

Question 24

Two nonzero vectors are orthogonal if their dot product is zero.

Question 25

Since \(f\) is linear, the maximum must occur on one of the vertices of the feasible region.

Question 26

A local maximum can only occur when \(f’\) is zero or undefined. Be careful that you check the domain of each branch.

Question 27

Use the Chain Rule.

Question 28

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

Question 29

Use integration by parts.

Question 30

Any basis for \(\mathbb{R}^n\) must consist of exactly \(n\) nonzero linearly independent basis vectors.

Question 31

Use the Rational Roots Theorem.

Question 32

What’s the relationship between the number of arrangements that have Pat ahead of Lynn and the number that don’t?

Question 33

Finding \(\text{lcm}(30,16)\) will come in handy.

Question 34

Think of a function with a horizontal asymptote of \(y=1\).

Question 35

The tangent plane to \(z=f(x,y)\) has normal vector \((f_x,f_y,-1)\). Alternatively, if you solve the answers for \(z\), the gradient of the surface has to equal the gradient of the tangent plane.

Question 36

Put the first four numbers in order and try different values of \(x\).

Question 37

Cancel out one thing immediately, and notice that \(k=(k-1)+1\).

Question 38

Sketching the graph of each function will go much more quickly than actually integrating.

Question 39

Answer choices A-C are different kinds of rectangular sums. Note that \(e^{-x}\) is decreasing.

Question 40

Notice that \(\text{Pr}(H>T)+\text{Pr}(H=T)+\text{Pr}(H<T)=1\).

Question 41

For \(f\) to have a relative maximum, you need \(\nabla f=0\), \(\det H_f>0\) (where \(H_f=\begin{pmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{pmatrix}\) is the Hessian matrix), and \(f_xx<0\).

Question 42

Sketch the points and think of translating the vectors between those points.

Question 43

Represent \(\begin{pmatrix}6\\7\\8\end{pmatrix}\) as a linear combination of \(\begin{pmatrix}0\\1\\2\end{pmatrix}\) and \(\begin{pmatrix}3\\4\\5\end{pmatrix}\). (The \(0\) helps.) Then use linearity.

Question 44

Use logarithmic differentiation.

Question 45

Check the units to make sure that you get gallons. (Remember that \(dt\) has a unit too!)

Question 46

You could find the characteristic equation, but the trace makes this so much easier.

Question 47

Sketch the region \([0,1]\times [0,1]\), and shade the region where \(|y-x|<\dfrac{1}{2}\).

Question 48

Look at the potential ranges of \(u\) and \(v\). Then figure out where the boundaries of \([0,1]\times [0,1]\) map to under the change of variables.

Question 49

You can either use substitutions and integration rules, or you can think of these geometrically (i.e. transformations).

Question 50

Note that if \(y^2=x^2\), then \(|y|=|x|\). (Why?) Think about how many ways the branches can be arranged.

Question 51

Use the Ratio Test.

Question 52

Use row reduction. Note that the system is homogeneous, so think of easy solutions.

Question 53

Sketch the circle, and use residue integration: \(\displaystyle\oint_C f(z)\,dz=2\pi i\cdot \displaystyle\sum_{i=1}^n \text{Res}(f,z_i)\) , where each \(z_i\) is a pole of \(f\) enclosed within \(C\).

Question 54

Set up the differential equation, and note that at in the limit, \(h'(t)=0\).

Question 55

Sketch a tiny bit of the curve at \(x=0\), paying attention to slope and concavity, and then use the fact that \(f^{\prime\prime}\) is increasing without bound to sketch the rest of \(f\).

Question 56

A metric must be positive definite (i.e. nonnegative, and zero only when two points are the same), symmetric, and satisfy the triangle inequality. To test the triangle inequality, try some simple points.

Question 57

Subrings must be closed under addition, subtraction, and multiplication.

Question 58

Try sketching a few functions over \((0,1)\), and remember that continuous functions preserve connectedness and compactness.

Question 59

Exactly two of the elements in the set must be equal. Think about what must equal the identity, and keep Lagrange’s Theorem in mind: the order of an element must divide the order of a group.

Question 60

Try replacing \(s\) with \(s+s\) or \(s+t\) and playing with the algebra.

Question 61

Factor the expression, and remember that \(p\) must be an odd prime. Also use Fermat’s Little Theorem: If \(p\) does not divide \(a\), then \(a^{p-1}\equiv 1\pmod{⁡p}\).

Question 62

Just focus on the terms that will multiply to give you \(x^3\): \(x^3\cdot x^0\) and \(x^1\cdot x^2\). Use the Binomial Theorem to get the coefficients.

Question 63

Sketch the graphs, but remember that exponential functions always eventually overtake polynomial functions.

Question 64

Continuous functions on compact sets are uniformly continuous. However, Lipschitz continuity is stronger than uniform continuity.

Question 65

Write \(p\) in the form \((x-r_1)(x-r_2)(x-r_3)\) and reason from there with the given information about \(p\) and \(p’\).

Question 66

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

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