Mathematical Biology 4010/6010 Fall 2012
Research Project: Smallpox Max points: 100 Due Tuesday, November 27
An outbreak of smallpox in Abakaliki in southeastern Nigeria in 1967 was reported by two
researchers Bailey and Thomas. People living there
1. a. (10 points) Use a Riemann sum (midpoint formula) with n : 3 partitions to approximate the area
. 7 1
under the curve f (x) = x2 on the mterval [1, 7]. answer: Ax = T = 2, ana1 so
area under curve ; [f(2)+ f(4)+ f(6) *2 = [(2)2 + (4)2 + (6)2 J* 2 = 5
A. Find the area of R.
The picture looks like this:
The points of intersection of the two curves are (0,1) and (1,0).
The top curve is the parabola y : 1 3:2 and the bottom curve is the line
3; : 1 I, so
No calculators allowed. Show your work.
1. (2 pts.) Find the supremum/maximum/inmum/minimum (write DNE if a number does not exist) of each of the
following subsets of R. Recall: The number sup A (inf A) is called the maximum (minimum) of A if sup A E A (i
THE COMPLEX EXPONENTIAL FUNCTION
(These notes assume you are already familiar with the basic properties of complex
We make the following denition
This formula is called Eulers Formula. In order to justify this use of
/ sin3 d3
(HINT: consider writing sin3 3 : sin3 sin2 3 and then use the identity cos2 3 +
sin2 3 : 1.)
SOLUTION: We have:
/ sing (3) d3 : f sin(3)(1 i cos2(3) d3
where for the last one we used the substitution u :
1. (a) (5 pts.) Give the denition of a nite set and of a countably innite set.
(b) (10 pts.) Show that if E Q R is a bounded nite set then sup E E E. (Hint: You may
want to use the Approximation Property for suprema.)
(c) (5 pts.) Give an example of an in
SHGW ALL WGRKII i
1. (5pm) Below are for directiens fields Circle the direction dd that gags with the
differential equatien y' : 4+1,
*7. f: .6,» ow mm»
'tu a ,l._'r',f/w.~_\
x, .t_ . \ _ d4: ,_/ ,«r 7
Problem 1.vLet R be the region bounded by the curves 9: = y2 and y = 53.
A. Find the volume of the solid generated by revolving the region R around the
Heres the picture of the region R:
0.5 - / /
032 o.'4 0'6 0'8 '1 132 1.'4
Applications to Physics (Motion of a mass on a spring)
What does it mean for a sequence to be increasing or decreasing? How do you show
that a sequence is increasing or decreasing?
Geometric Series, Pseries, Harmonic Series
Final Exam Review Math 131 L. Ballou
1. If f(x)= , nd What is the domain of
The function, is not defined at X : 0, and the
composite is not dened at X I 3, so the domain is all reals not equal to 0 or 3.
2. Are the lines 2x + y =1 and 2x y =1 pe