Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: | (208) |
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**Problem 1. ** Suppose that is nondecreasing on .
Notice that is automatically bounded on , because
for any in .

- If is a partition of , write formulas for and in as simple terms as possible.
- Suppose that for each . Show that .
- Prove that is integrable.

**Solution. ** Notice that as is nondecreasing,
and
. Hence,

- The upper and lower sums are given by
- If then and likewise . Hence . By telescopic summation, the last sum is .
- Setting we get partitions with . This can be made arbitrariry small, and hence can be made arbitrarily small, hence is integrable.

**Problem 2. ** In each of the following, is a
continuous function on .

- Show that
- Characterize the functions that have the property that
for all .

**Solution. **

- If is a primitive of , i.e., , then
. Hence using the
second fundamental theorem of calculus,

- Differentiate both sides of the equality using the first fundamental theorem of calculus and get that and thus that for all .

**Problem 3. **

- Prove that if two functions and both satisfy the differential equation and if they have the same value and the same first derivative at 0, then they are equal.
- Use the above to show that for all . (Do not use the formula for the of a sum!)

**Solution. **

- Set . As , it follows that and clearly . Now set and calculate . Hence is a constant. But hence is the constant 0. But as the two terms composing a non-negative, this forces each of them to be 0, and in particular and thus .
- Set and . Then , , and . Hence and both satisfy the differential equation . In addition, and . Hence the requirements for the theorem of the previous part of this problem are met, and hence its conslusion is satisfied. So or .

**Problem 4. **

- Compute
- Use your result to estimate the difference between and . Warning: a 10 digit answer obtained with your calculator may contribute negatively to your grade. You shouldn't use any calculating device and your derivation of the answer should be simple enough that it be clear that you didn't need any machine help.

**Solution. **

- Using L'Hôpital's rule twice, we get
- From the above we get that is approximately if is small. Hoping that is small enough, we find that , meaning that .

**Problem 5. ** Evaluate the following integrals in terms of
elementary functions:

- (cancelled)

**Solution. **

- We integrate by parts twice:

- Not required, though still, .
- Take and then
and hence
and
so

- Take and then and so

**The results. ** 83 students took the exam; the average grade
is 65.31, the median is 65 and the standard deviation is 25.17.

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

Dror Bar-Natan 2003-02-17