Problems

1. 1. In the course of Activity 1 we also computed the area of the surface of the earth. Where did we do that? What is it?
   2. How is the volume of the crust related to the area of the surface? Is this reasonable? Why or why not?

   For more about the Earth, visit Earth.

2. Suppose we were able to determine that the (average) radius of the earth is actually 3959 miles, plus or minus 1 mile.

   1. Calculate the area of the earth's surface.
   2. How far off might this calculation be if our radius is actually off by 1 mile?
   3. Find some region (city, state, country, continent) whose area is approximately equal to the possible error.
   4. Calculate the percentage error. Is this error small or large?

   For facts about the Earth's radius, visit Earth Radius.

3. There is a technical problem with the way we described “differential” in that the word has two definitions depending on whether we are talking about independent or dependent variables. The same variable sometimes plays both roles in a single context — for example, the intermediate variable`u` that we introduced in our discussion of the Chain Rule. How are we to know that these possibly different meanings are actually the same?
1. First consider the case of the function `y=t`, that is, the function for which independent and dependent variables always have the same value. Sketch the graph of this function. Explain why `dy=dt` in this case.
2. Now consider a situation in which is `y` a function of `u` and `u` is a function of `t`. Thus `u` is both an independent and a dependent variable. Explain why the Chain Rule requires that `du` means the same thing whether `u` is viewed as independent or dependent.