M161, Final Exam, Spring 2003
You may use alulators. You are not allowed to have information stored in your alulator.
When you are told to do a omputation analytially, you must show all of your work. When
you are told to do a problem analytially, you will get no points for alulator results.
1. (a) Find the 4th order Taylor polynomial approximation of the funtion f (x) = os(x)
about the point a = =4.
(b) Use Taylor's Inequality (jRn (x)j nM jx ajn ) to estimate the auray of the
approximation f (x) T (x) when x satises 0 x =2.
+1
(
+1)!
4
() Find the seond order Taylor polynomial approximation of the funtion f (x) =
about the point a = 0.
1
1 x
p
2. For this problem
p you must show your work. Calulator results are not suÆient.
(a) Find (1
3i) .
5
(b) Express e
i
2+
in the form a + ib.
1 + 3i
in the form a + ib.
2 + 5i
() Express
(d) If z and w are omplex numbers, show that
3. Calulate the following integrals.
Z
px ln x dx
(a)
be integrated analytially.
z w
z
=
.
w jwj
2
You must show your work. These integrals must
If you just give the result from your alulator, you
will get zero redit.
Z
(b)
Z
()
x
2
5
2x
1
t +t
3
dx
6
2
dt
4. Evaluate the following limits. Show how you get your answer. A plot of the funtion is
not suÆient (even though it may be helpful).
x 3x + 2
(a) xlim
.
!
x 1
2
1
2
(se x
(b) lim
x! 2
tan x).
5. For the urve given by the parametri equations x = t(9
4 t 4:
1
t ), y = (t
2
2
1)(t
2
9),
(a) Sketh the urve and indiate with an arrow the diretion in whih the urve is traed as
the parameter t inreases.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−3
−2
−1
0
x
1
2
3
(b) Loate all values of t 2 ( 4; 4) where the urve has either a horizontal or a vertial
tangent (you must show work|piking them o of your alulator is not suÆient).
6. (a) Evaluate the integral
Z
1
dx as a power series about a = 0.
1+x
4
x
about a = 0 and determine
(b) Find a power series expansion for the funtion f (x) =
4x + 1
its interval of onvergene.
7. Find the length of the urve y = ln(os x), 0 x =4. (Suggestion: Use your alulator
to evaluate the integral.)
8. Find the radius of onvergene and the interval of onvergene of the series
2
X1 n(x + 2) .
n
n=0
3n
+1