2011-2012 The motion of a grizzly bear stalking its prey, walking left and right of a fixed point in feet per second, can be modeled by the motion of a particle moving left and right along the x-axis, according to the following acceleration equation: Assume that the origin corresponds to the fixed point, and that … Continue reading "Problem 15: A Grizzly Motion Problem"
2011-2012 Find all angles on the interval at which the tangent line to the graph of the polar equation is horizontal. Solution: Express the polar equations parametrically (in terms ofx and y) and calculate the slope of the polar equation. The tangent lines to the polar graph are horizontal when the numerator of this derivative is … Continue reading "Problem 13: Polar Derivatives"
2011-2012 Have you ever loved something so deeply, so meaningfully, so completely, so profoundly that it would really irk you if you dropped that thing into a bubbling vat of acid? I have, and so that you may learn from my tragedy, I will share a horrific tale from my past. Once, on a whim, … Continue reading "Problem 12: Super Related Rates"
2011-2012 Chain Rule Problems Calculate the derivative of tan2(2x –1) with respect tox using the chain rule, and then verify your answer using a second differentiation technique. Solution for Chain Rule Practice Problems: Note that tan2(2x –1) = [tan (2x – 1)]2. To find the solution for chain rule problems, complete these steps: Apply the power rule, … Continue reading "Problem 11: Chain Rule"
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)"