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1.
The representation is a finite continued fraction, and probably the easiest way to differentiate it is to “unwind” it to a normal fraction so num(x)/denom(x) and differentiate that as usual. Continued fractions (infinite) can be used to make the link between algebraic irrational numbers and the corresponding integer coefficient algebraic equations and analyse transcendental irrational numbers like “e”
2.
Can solve it properly using average speed or harmonic mean inequalities, or a simple problem solving way would be to consider a particular headwind which would make the difference clear (assuming the result was then valid for all headwinds). The AM-GM-HM inequalities are relatively famous and have a nice visual interpretation in terms of a semicircle.
3.
The function is y*f(x) = 1 and its derivative y’/y = - f’/f, so 1/x, 1/sin(x), 1/x^2, … Reason about their signs, evaluate f(x) at certain x values, where f=0,1, reason about its asymptotic behaviour and similar for derivatives ...
4.
Fairly standard definite integral, though a quick sketch or local taylor/maclaurin series would highlight that the integrand is approximately quadratic so roughly ex^2 from (0,0) to (1,e). So the definite integral will be ~e/4 so about 0.7. So anything too different from this should be checked.
5.
Fairly straightforward odd / dots argument. Its well known that odd^2 = 1 mod 8, so the question is just asking that, so there are a few ways to go. Squares and mod 3 came up in the "extra mat" recently.
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6. “Standard” (crops up on a few such lists) derivative question so y=x^x and take logs then implicit/chain rule .. .or write the rhs as e^ln(#) …
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7. How the previous argument for 6) can be adapted for x^x^x and made recursive/inductive for x^x^x^x …
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8. Well known examples correspond to trying to solve x^x^x^... = 2 or = 4. For the (infinite) tetration problem, as the previous examples should illustrate, convergence is important. A rough proof of the domain might correspond to finding the largest x for which the standard substitution is valid.
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9. Easy to google the basic problem, though worthwhile trying to analyse using conditional
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10. Think about factors (how you generate a number abcabc) or a remainder (mod) argument…
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11. Logistics map, could write it as a completed square to emphasise the shape of the parabola, though noting the roots and the gradient (linear with gradient -2k, zero at x=0.5) at the fixed points are things of interest here. Sketch the parabola and y=x line to iterate graphically. So relevant domain is (0,1) and k=4 maps that interval back to itself. What are fixed points, what happens when you start close to those values…. Sample k at 2, 4 say, and think about the different results.
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12. Think about the triangle inequalities for the diagonals.
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13. Think about adapting the usual sqrt(2) argument.
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14. Think about the number of 5s you have, and are the number of 2s relevant?
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15. Question a bit vague, but assuming its making the usual assumptions, choose the colour of the first portion then what about the other two? How many are repeated (rotations)? For 3 portions, no maths is really required, but what about if it was 10?
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16.Sketching an exponential and rotating a line which passes through the origin should be straightforward. A simple, but worth doing, calculation gives the value of k which is tangent to the curve so forms the boundary (1 solution) between 2 and 0 solutions.
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17.Simple mod proof
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18.Well known counter example which has some interesting properties.
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19.Question deals with exponentiation so taking logs is a natural thing to do and it can be rearranged in the form of the given expression. Sketch the given function, why are there 2 solutions (for a,b>1), what is the maximum of the given expression and what can you then say about the intervals for a and b, assuming a and b, assuming a<b.* If there is an integer solution, what must the value of a be and therefore can you determine b?* The solution to such equations can be expressed using the lambert W function, but this is outside elementary analysis, hence the exploratory / analysis nature of the question.* Note the similarity with Q8 for the log(x)/x part.
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*20.Expand as a geometric series (infinite tree) or think about a recursive way of representing p(A) (simple tree with a cycle) or think about a simple ratio argument for each turn. As should be expected, p(A) is slightly greater than ½ as A rolls first. Three players is similar,
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Uk university (mathematics w/statistics) + 40 Predicted Points in IB