Math402/Advanced Calculus I Test Study Guide

----Final Exam-----

The final exam is scheduled as follows:

     Tuesday, December 15, 1998 6:50PM-8:50PM HH204

    

....General Information About The Exam....

....(this information is the same as it was for Test #1 & #2)....

....(check the test #1 test guide page for details )....

A FEW IMPORTANT REMINDERS!

The test will be worth 120 points and will 
constitute roughly 35% of the total number of test points
for the course.

You will have 2 hours to work on the test.

All of your work must be placed in the "Blue Book(s)"
that will be provided for you.

The answers in the Blue Book(s) must appear in the same order
as they appear on the test sheet, and so you will need to
plan ahead and leave necessary space if you intend to
work the problems out of order.

    

....Specific Information About The Exam....

   The first part of the final will be worth 40 points and will
   cover topics included on the first two tests. To prepare for
   this part of the exam you should revisit those exams since
   quesions on this part of the final will be similar to the 
   questions on those exams.
   
   The second part of the final will be worth 80 points. 
    
0. Know how to work any assigned homework problem
   (from section 5.1 through  6.4)
   
1. Be able to prove L'Hospital's Rule (Th 4.20)

2. Be able to prove that if l(x) solves the ODE l'(x)=1/x & l(1)=0 
   for x > 0 then
  
                  l(ab)=l(a)+l(b)
                  l(a^r)=lf(a)    for r rational
                  l has an inverse function e
                  e(0)=1
                  e'(x)=e(x)  for x in R
                  e(a+b)=e(a)e(b)
                  (e(a))^r=e(ar)  for r rational

3. Be able to prove Lemma 5.6 
   That is, if   f''(x) + f(x) = 0 for x in R and f(0)=f'(0)=0
            then f(x)=0 for x in R

4. Be able to prove that 
   if   f''(x) + f(x) = 0 for x in R and f(0)=a & f'(0)=b
            then f is unique 

5. Assuming that s(x) solves the ODE in 4. with a=0 & b=1 and c(x)
   solves the ODE in 4. with a=1 and b=0, then
                   c(-x)=c(x)  for x in R
                   s(-x)=-s(x) for x in R
                   [s(x)]^2 + [c(x)]^2 = 1   for x in R
                   c(a+b)=c(a)c(b)-s(a)s(b)  for x in R
                   s(a+b)=s(a)c(b)+c(a)s(b)  for x in R

6. Be able to define Upper and Lower Darboux Sums                  
                   
7. Be able to define and Integrable Function 

8. Be able to define the First and Second Integrability Criteria (FIC & SIC)

9. Be able to prove f(x) is Integrable if and only if f satisfies 
   the FIC  
   
10. Be able to prove that if f:[a,b]->R is monotone then f is Integrable                     
   
11. Be able to prove that if f:[a,b]->R is continuous then f is Integrable 

12. Be able to prove The First Fundamental Theorem of Calculus  

13. Be able to define Reimann Sums; and assuming The Darboux Sum
    Convergence Criterion (Th 6.13) prove that if f(x) is Integrable
    on [a,b], and Pn is a sequence of partitions such that 
    Lim||Pn||=0 then LimR(f,Pn)= the definite integral of f from a to b    

(last updated 3:15PM on 12-9-98)