Gloria Y.

Tutor for 2+ years, undergraduate at Princeton University

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SAT II Mathematics Level 2

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Question:

The number of hours of daylight, $$d$$, in Hartsville can be modeled by $( d = \frac{35}{3} + \frac{7}{3} \sin \Big(\frac{2\pi}{365} t \Big), $) where $$t$$ is the number of days after March 21. The day with the greatest number of hours of daylight has how many more daylight hours than May 1?

Gloria Y.

Answer:

Clearly one must find the number of daylight hours on May 1, and the maximum number of daylight hours possible, and subtract these two numbers. Finding the number of daylight hours on May 1 simply requires finding the number $$t$$ of days after March 21 and plugging into the formula. Just be careful you're counting correctly! To find the maximum number of daylight hours possible, notice that this occurs when $$\sin \big(\frac{2\pi}{365} t \big)$$ is maximized, because that's the only part of the expression for $$d$$ that involves $$t$$. Recall that $$\sin(x)$$ reaches it's maximum of $$1$$ at $$x = \frac{\pi}{2}$$, so just substitute $$x$$ for $$\frac{2\pi}{365} t$$ and solve for $$t$$. Notice that the result is a non-integer for $$t$$, so you'll have to round $$t$$ to the nearest integer before calculating the resulting value of $$d$$, which will be the maximum number of daylight hours possible. Once you've done that, just do the final subtraction!

Java Programming

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Question:

How can I write a Java program that calculates the largest prime factor of the number $$600851475143$$?

Gloria Y.

Answer:

This question is tricky because $$600851475143$$ is very large. Notice first that it can't be stored in an int, so you should store it in a long. Here's one method, perhaps the most intuitive one. Create a champion that stores the largest prime factor you've found so far. Use an algorithm like Euler's sieve to store primes up to half of $$600851475143$$ in an array (you'll need to decide how to do this, but it shouldn't be too bad). Use a for loop to go through all the integers from $$1$$ to half of $$600851475143$$, and check divisibility only if you're looking at a prime number (why only half of $$600851475143$$?). If you find a prime that divides $$600851475143$$, update the champion. (What does it mean if you don't find a prime that divides $$600851475143$$ in this for loop?) Here's another method, which might speed things up. Suppose you find a prime $$p$$ that divides $$600851475143$$. Then any other (larger) prime that divides $$p$$ is going to divide $$q=600851475143/p$$. This suggests that you can do your testing on the smaller quantity $$q$$ instead. So, if you can create a method that finds the first prime factor of a number, then you can call it recursively on smaller and smaller numbers using this idea.

Calculus

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Question:

The plane $$y = 1$$ slices the surface $( z = \arctan \frac{x+y}{1-xy} $) in a curve, $$C$$. Find the slope of the tangent line to $$C$$ at the point where $$x = 2$$.

Gloria Y.

Answer:

Your first thought when you see the phrase "find the slope of the tangent line" should be, "I need to calculate a derivative of some sort." The question is, "what derivative?" Consider the fact that you are looking at a curve $$C$$ for which the $$y$$-coordinate is always 1. This suggests that you should hold $$y$$ constant in the formula $$ z = \arctan \frac{x+y}{1-xy} $$. That means you should differentiate $$z$$ with respect to $$x$$. Then, of course, evaluate your derivative at the point where $$x = 2$$.

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