Term Exam 4
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:
Prove the following:
- 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
Problem 2. Compute the following integrals: (a few lines of
justification are expected in each case, not just the end result.)
. (This, of course, is
- 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.
- 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
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 ,
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