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Section Assignment #8 - Done In Class on Thursday 3/13

All of these questions have to do with continuity.
  1. Let \(P(t)\) be the cost of parking in a New York City parking garage for \(t\) hours. The policy of the garage is that \(P(t)\) is $20 per hour or fraction thereof. For example, if you are in the garage for two hours and one minute, you pay $60.
    If \(t_0\) closely approximates some time \(T\), then \(P(t_0)\) closely approximates \(P(T)\).
    True or false? Justify your answer.

    2. This is discretely asking whether or not you think the function \(P(t)\) is continuous. Remember, one way to think about what it means for a function \(f(x)\) to be continuous is that small change in \(x\) results in a small change in \(f(x)\). That is not the case here. \(P(t)\) has a jump at every integer. For example, \(P(t)=20\) for \(t \in (0,1]\), but \(P(t)=40\) for \(t\in (1,2]\). So if \(T=1\) and \(t_0=1.01\), then \(t_0\) is close to \(T\), but \(P(t_0)\) is not close to \(P(T)\).
  2. You decide to estimate \(e^2\) by squaring longer decimal approximations of \(e=2.71828...\).
    1. This is a good idea because \(e\) is a rational number.
    2. This is a good idea because \(y=x^2\) is a continuous function.
    3. This is a bad idea because \(e\) is irrational.
    4. This is a good idea because \(y=e^x\) is a continuous function.
    Clearly explain your choice

    3. The correct answer is b.
    The prompt says that we are making slight changes to \(x\). For example, first we consider \(x=2.7\), then \(x=2.71\), then \(x=2.718\), etc, and we square each of these values. That is, we are making small changes to \(x\) before we plug into the function \(f(x)=x^2\). Since we know \(f(x)=x^2\) is a continuous function, small changes in \(x\) lead to small changes in \(f(x)\), AND, we know that \(\lim_{x\rightarrow c}f(x)=f(c)\), so these small changes are getting us closer and closer to the true value of \(f(e)=e^2\).
  3. At some time since you were born, your weight in pounds equaled your height in inches.
    Do you think this is true or false, and why?

    4. If you agree that your height and weight are continuous functions of time, this statement is true.
    As many of you stated in class, at birth your weight is less than your height in inches, but now, your weight is of course much greater than your height in inches. If weight and height are continuous, since \(w(0)-h(0)<0\) and \(w(today)-h(today)>0\), there must be some point in time between when you were born and today where \(w(t)-h(t)=0\) by the Intermediate Value Theorem. This implies there was some time where your weight and height in inches were exactly the same.