Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (27) |
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this document in PDF: Solution.pdf

**Problem 1. **
All that is known about the angle is that
. Can you find
and
? Explain your reasoning in full detail.

**Solution. ** (Graded by C.-N. (J.) Hung)
In class we wrote the formula
. Also
using
and taking
we get

**Problem 2. **

- State the definition of the natural numbers.
- Prove that every natural number has the property that whenever is natural, so is .

**Solution. ** (Graded by V. Tipu)

- The set of natural numbers
is the smallest set of numbers
for which
- ,
- if then .

- ,
- if then .

- Let be the assertion ``whenever is natural, so is ''.
We prove by induction on :
- asserts that ``whenever is natural, so is ''. This is true by the second bullet in the definition of .
- Assume , that is, assume that whenever is natural, so is . Let be a natural number. Then is a natural number because by the number is natural and because adding one to a natural number gives a natural number by the second bullet in the definition of . So we have shown that whenever is natural so is , and this is the assertion .

**Problem 3. **
Recall that a function is called ``even'' if
for all
and ``odd'' if
for all , and let be some
arbitrary function.

- Find an even function and an odd function so that .
- Show that if where and are even and and are odd, then and .

**Solution. ** (Graded by C. Ivanescu)

- Set
and
. Then
while
(so is even)
and
(so is
odd).
- Assume where is even and is odd. Then
*necessarily*. Now if as above, then both and can play the role of in this argument, so they are both equal to and in particular they equal each other. Likewise,

**Problem 4. **
Sketch, to the best of your understanding, the graph of the function

**Solution. ** (Graded by C. Ivanescu)

If then and so ; furthermore, the larger is (while ), the larger is and hence the smaller is. When approaches from above, approaches 0 from above and hence becomes larger and larger. If the and so . When , and when approaches from below, approaches from below and approaches 0 from below, and so becomes more and more negative. In summary, the graph looks something like:

**Problem 5. **

- Suppose that for all , and that the limits and both exist. Prove that .
- Suppose that for all , and that the limits and both exist. Is it always true that ? (If you think it's always true, write a proof. If you think it isn't always true, provide a counterexample).

**Solution. ** (Graded by C.-N. (J.) Hung)

- Let
and
and assume by contradiction that ; that is, that
. Use the existence of the two limits to find
and
so that
- Take for all and for all and
. Then for all but
. So it isn't always true that if
for all and the limits exist, then
.

**The results. ** 105 students took the exam; the average
grade was 67.19, the median was 70 and the standard deviation
was 21.12.

The generation of this document was assisted by
L^{A}TEX2`HTML`.

Dror Bar-Natan 2004-10-18