If a + bi = (3 + i)/(1 + i), find the values of a and b.
There are two valid ways to solve this problem. Most students will multiply the right-hand-side by (1 - i)/(1 - i). If done correctly, when the top binomial is FOILed, this will give a + bi = (4 - 2i) / 2 = 2 - i, and therefore a = 2 and b = -1. Be careful: when we get an i^2, it becomes NEGATIVE 1. The other way to solve this is by multiplying both sides by (1-i). This gives a - b + ai + bi = 3 - i, which means a - b = 3 and a + b = 1. Solving this system gives a = 2 and b = -1. Both skills (multiplying by the conjugate, and deriving a system of equations) are important for other questions on the Math SAT II
In an experiment, you burn pure magnesium in the open air to determine the empirical formula of magnesium oxide. You determine the empirical formula by comparing the mass of the pure magnesium to that of the final compound. However, you suspect that some nitrogen in the air also reacted with the magnesium. How would this error affect your result?
Nitrogen will react with magnesium to form magnesium nitride. At first, many students will reason that this is an impurity that adds extra mass. Since that extra mass increases the final math, many students reason that this will increase the calculated ratio of oxygen to magnesium. However, the correct answer is more complicated. When nitrogen reacts with magnesium, two nitrogen atoms of 14.0 amu each will bond with 3 magnesium atoms, adding mass of 28.0 total amu or 9.3 amu per magnesium atom. Had oxygen reacted instead, one oxygen atom of 16.0 amu would have reacted with each magnesium atom. Therefore, the nitrogen is actually lowering the final mass of the compound, causing the scientist to calculate a ratio of oxygen to magnesium that is too low.
Please draw the function f(x) = x^3 - 4x, its first derivative, and its second derivative. Label all x and y intercepts, but do not worry about scale.
f(x) is a typical cubic that comes from (-inf, -inf), crosses the x-axis at -2, 0, and 2, then goes off to (+inf, +inf). The first derivative f'(x) = 3x^2 - 4 is an upward-opening parabola that intersects the x-axis at x = +/- 1.15, and whose vertex is at (0,-4) The second derivative f''(x) = 6x is an steep upward sloping line that passes through the origin.