Kurt V.

High School Math Department Chair

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Pre-Calculus

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Question:

Convert the point (-2, 2) from rectangular (cartesian) to polar form (r, theta).

Kurt V.

Answer:

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).

Calculus

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Question:

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.

Kurt V.

Answer:

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.

Statistics

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Question:

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?

Kurt V.

Answer:

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%).

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