2011-2012 For the second year in a row, one toy looks to dominate the market once again: My First Cliff Diving Kit. It is all the rage, because it comes with everything you need to be an effective cliff diver: swim trunks, neck brace, legal documents for naming next of kin, and very detailed one-paragraph … Continue reading "Problem 10: Diving into Rates of Change"
2011-2012 Derivative Graph In the below graph, two functions are pictured, f(x) and its derivative, but I can’t seem to tell which is which. According to those graphs, which is greater: Solution: You might have noticed that the red function has an even degree whereas the blue has an odd degree. Why? An even-degreed function’s ends will … Continue reading "Problem 9: The Graph of a Derivative"
2011-2012 A man wants to build a rectangular enclosure for his herd. He only has $900 to spend on the fence and wants the largest size for his money. He plans to build the pen along the river on his property, so he does not have to put a fence on that side. The side … Continue reading "Problem 8: Optimizing a Dirt Farm"
2011-2012 Four functions are continuous and differentiable for all real numbers, and some of their values (and the values of their derivatives) are presented in the below table: If you know that h(x) = f(x) · g(x) and j(x) = g(f(x)), fill in the correct numbers for each blank value in the table. Solution: Woo, … Continue reading "Problem 7: Revenge of Table Derivatives"
The position of a particle (in inches) moving …
2011-2012 During a taping for Circus of the Stars, beloved actress Betty White is shot out of a cannon. The firing goes completely awry and sends her on a collision course with a jet. As they converge, Betty and the jet plane at right angles to each other (see diagram below). Betty is 200 miles … Continue reading "Problem 4: B-B-B-Betty and the Jet (Related Rates)"
2011-2012 Let f(x) be the function defined below: Determine whether f(x) is continuous at x = 0 and explain your answer. Note: You may use a graphing calculator to examine the graph of f(x). Solution: If f(x) if continuous at x = 0, its left- and right-hand limits exist at x = 0, and they … Continue reading "Problem 2: Continuity"
2011-2012 Describe or draw a function, f(x), with the following characteristics: f(x) has domain (–∞,8) f(x) has range (–∞,9) f(4) = 0; f(5) = 0; f(7) = 0 The limit, as x approaches –∞, of f(x) equals 9 The limit, as x approaches 8 from the left, of f(x) equals –∞ f(–1) = f(–3); f(–1) … Continue reading "Problem 1: Limits"
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