Convert the point (-2, 2) from rectangular (cartesian) to polar form (r, theta).
Using knowledge of trigonometry, we can plot the ordered pair and create a right triangle. Then, pythagorean theorem provides r^2 = (-2)^2 + (2)^2. Thus r^2 = 8 and r = sqrt(8) or 2*sqrt(2). We can then identify the theta value using inverse tangent of 2/-2 or tan inverse(-1). This indicates that the theta value is 135 degrees since the point must lie in Quadrant 2. Therefore the ordered pair (-2, 2) can be expressed in polar form as (2sqrt(2), 135 degrees).
Use the second derivative test to determine the intervals over which the polynomial f(x) is increasing, decreasing, or constant. Use f(x) = x^2 + 12x - 6.
The first derivative of f(x) is f'(x) = 2x + 12. The second derivative of f(x) is f''(x) = 2. Since f''(x) > 0 for all values of x between negative infinity and infinity, the function f(x) is increasing over its entire domain.
A group of 11 students (7 boys, 4 girls) is selected at random to form a 4 person committee. What is the probability that the committee contains at least 1 girl?
First, we know that the order in which students are selected does not matter, so we use Combinations rather than Permutations. Next, we can find the probability of selecting a committee with 0 girls. That is found as 7C4/11C4 = 35/330 = .10606. Since all other possible committees include at least 1 girl, the probability of that happening is 100% - 10.61% or 1 - .10606 = .8939 (89.4%).