Mathematical Biology 4010/6010 Fall 2012
Research Project: Smallpox Max points: 100 Due Tuesday, November 27
Name:
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.
Answer:
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
142/0 [(1I2)(133)]da:
1
2/ [1I21+a:]d55
0
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
Math 307
THE COMPLEX EXPONENTIAL FUNCTION
(These notes assume you are already familiar with the basic properties of complex
numbers.)
We make the following denition
cw:cos9+zsin9. (1)
This formula is called Eulers Formula. In order to justify this use of
1. Find
/ 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
/sin3d3/sin3cos23d3
=icos3+/u2du.
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,
yhm
"'f Il"~-~x\
dww /.«r_m\
*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
arc-axis.
Answer:
Heres the picture of the region R:
1.5- /
0.5 - / /
032 o.'4 0'6 0'8 '1 132 1.'4
Applications to Physics (Motion of a mass on a spring)
Sequences
What does it mean for a sequence to be increasing or decreasing? How do you show
that a sequence is increasing or decreasing?
Innite Series
Geometric Series, Pseries, Harmonic Series
Integra
Final Exam Review Math 131 L. Ballou
3x
x
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