© | Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I: | (101) |
Next: Solution of Term Exam 4
Previous: Class Notes for Thursday March 17, 2005 |

This document in PDF: TE4.pdf

University of Toronto, March 21, 2005

**Solve all of the following 4 problems. ** Each problem is
worth 25 points. You have an hour and 50 minutes. Write your answers in
the Term Exam notebooks provided and **not** on this page.

**Allowed Material: ** Any calculating device that is not
capable of displaying text.

**Problem 1. ** Agents of
CSIS have secretly developed a
function that has the following properties:

- for all .
- is differentiable at 0 and .

- is everywhere differentiable and .
- for all . The only lemma you may assume is that if a function satisfies for all then is a constant function.

**Problem 2. ** Compute the following integrals: (a few lines of
justification are expected in each case, not just the end result.)

- .
- (assume that and that and ).
- .
- . (This, of course, is ).

**Problem 3. **

- State (without proof) the formula for the surface area of an object defined by spinning the graph of a function (for ) around the axis.
- Compute the surface area of a sphere of radius 1.

**Problem 4. **

- State and prove the remainder formula for Taylor polynomials (it is sufficient to discuss just one form for the remainder, no need to mention all the available forms).
- It is well known (and need not be reproven here) that the th
Taylor polynomial
of around 0 is given by
. It is also well known (and need not be
reproven here) that factorials grow faster than exponentials, so for any
fixed we have
. Show that for large enough ,

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

Dror Bar-Natan 2005-03-23