7.1 Integration by parts
Important derivatives (memorize them):
dim : Turn1
E
% lna: : i, but be aware of the absolute value sign in d2: : ln + 0.
i870 : ex
dx
id sina: : cosa:
if
7d cosx 7 *SIHSE
x
d
rst
Please put all work and solutions in the stamped blue book provided. Begin each new
problem on a new page (the back ofthe sheet is OK). Put your name, form of test
(A or B), row number, and seat n
l. (6 pts. each) Consider the function f(z) : 82 : ewfy.
(a) Find real functions u(a:,y) and v(m,y) such that f(2: + iy) : Margy) + efgy).
(b) ls f(z) analytic? Justify your answer using the CauchyRie
1. (a)13?9:(274)+(1 (2mm 2)E 25:3j+5k
(b) F:(472t)i+(72+3t)j+(2+5t)h
(c) Wehave
55242t
y:2+3t
z:2+5t
(d) We plug the 53, y and z from above (a point on the line) into 23: 3y + 102 = 3:
2(4 2t) 3( 2:3t
Linear systems of rst order DEs:
x:A(t)x+g(t)
Initial conditions: x(t0):x0
Existence/uniqueness theorem: The Initial Value Problem x:A(t)x+g(t), x(t0):x0 has a
unique solution in the largest open inte
Chapter 5 Integrals
In Chapter 2 we used the tangent and velocity problems to introduce the derivative,
which is the central idea in differential calculus. In much the same way, this chapter
starts wi
MATH-4435, Spring-2016
Test #1
Name:
MATH-4435/6435 Linear-II
Test-1
Directions: Clearly show your work. For full credit, all your answers must be justified by
your work.
1. (8 points each) For each