OSH/2

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  1. You are planning to drive along a limited-access highway and want to be prepared for fuelling up in case your fuel tank runs low. Below is a map of the part of the road in which you expect to be running low on gas. Note: a limited-access highway is a highway which you can only get on/off at entrance/exit ramps and there is no option of making U-turns between exits.
    1. As a warm-up exercise, consider the function $d(x)$ that gives the distance from a point on the x-axis $(x, 0)$ to the point $(2,3)$. Write down an expression for $d(x)$ and plot it. Use asymptotics to make sure that you get the correct behaviour of the function as $x \to \infty$ and $x \to -\infty$.
    2. Using the distances in the map below, consider the function $d_{crow}(x)$ that gives the distance from a point on the highway $x$ km from the start of the drive ($x=0$) to the nearest gas station assuming that you are in a 4x4 vehicle that can drive off-road in a straight line from the point $x$ to the nearest gas station. Plot $d_{crow}(x)$. You do not have to write down an expression for this function and the plot does not have to be precise (let's call it a sketch). However, the scales on the axes should be realistic and important features (in particular, the x and d coordinates of minima and maxima) should be clearly marked and accurate. Note: the subscript "crow" is a reference to the idiom "as the crow flies".
    3. Is $d_{crow}(x)$ a continuous function? Explain your answer, in particular, any claims about continuity (or lack thereof) at locations where the nearest gas station changes.
    4. Is $d_{crow}(x)$ a differentiable function? Explain your answer, making reference to any/all points at which the function is not differentiable.
    5. Again using the map below, consider the function $d_{drive}(x)$ that gives the distance from a point on the highway $x$ km from the start of the drive to the nearest gas station assuming that you are following the highway and roads to the nearest gas station according to the usual laws and traditions. Plot $d_{drive}(x)$. As before, you do not have to write down an expression for this function and the plot does not have to be precise (let's call it a sketch). However, the scales on the axes should be realistic and important features (in particular, the x and d coordinates of minima and maxima) should be clearly marked and accurate. You can assume that all ramps are short enough that the distance driven along them can be ignored.
    6. Is $d_{drive}(x)$ a continuous function? Explain your answer, in particular, any claims about continuity (or lack thereof) at locations where the nearest gas station changes.
    7. Is $d_{drive}(x)$ a differentiable function? Explain your answer, making reference to any/all points at which the function is not differentiable.
    A map of the section of highway discussed in Problem 1.
  2. Bears search for berries that grow in patches that can be spread out across a large area. A bear will spend time in one patch gathering food before moving to another patch. The number of berries collected in a patch depends on the amount of time spent in the patch: $$B(t)=\frac{At}{k+t}$$ where $B(t)$ is the number of berries collected at the end of $t$ units of time spent in the patch. The constants $A$ and $k$ are positive. Consider that as a bear spends more time in a patch and collects the available berries, it becomes more difficult to find the remaining berries. This feature is known as diminishing returns.
    1. Using the definition of the derivative, calculate $\frac{dB}{dt}$ for arbitrary $t$. Your answer should depend on the unspecified constants $A$ and $k$.
    2. For what values of $t$ is this derivative positive? negative? What does this mean in terms of the bear's berry-picking efforts?
    3. Relate the difficulty of finding berries to the derivative of $B(t)$. That is, what trend in the derivative would indicate that it is getting harder to find berries?

See the related example if you are not sure what is expected of you in terms of "good mathematical communication".