7.1 Integration by parts
Important derivatives (memorize them):
dim : Turn1
% lna: : i, but be aware of the absolute value sign in d2: : ln + 0.
i870 : ex
id sina: : cosa:
7d cosx 7 *SIHSE
d1 tanx : sec2 x
i - i1 7 1
dm Sin 3 i 1_x2
i 1 n _ 1
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 number on the outside ofthe blue book (preferably
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 CauchyRiemann equations.
mié'y m iiy _
e = e e em
1. (a)13?9:(274)+(1 (2mm 2)E 25:3j+5k
(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):10(2:5t)*0
8 42516 9t120:50t* 3
Linear systems of rst order DEs:
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 interval I in t that contains to for which A(t) and g(t)
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 with the area and distance problems and uses them to form
Directions: Clearly show your work. For full credit, all your answers must be justified by
1. (8 points each) For each these transformations on the vector space of polynomial