University of California, Riverside
Department of Mathematics
Midterm
Mathematics 9C - First Year of Calculus
Sample 1
Instructions: This exam has a total of 60 points. You have 50 minutes.
You must show all your work to receive full credit You may use an
9C - 13.7 - THE ROOT TEST AND THE RATIO TEST - FALL
2013
PAOLO MANTERO
Last time: we have seen Absolute Convergence/Conditional Convergence
Let me recall you the Absolute Convergence Test.
P
Theorem 1 (Absolute Convergence Test). If
an is absolutely conve
MATH 09C 13.9 CALCULUS WITH POWER SERIES
PAOLO MANTERO
Recall: Last time we have seen power series centred at a. Today, we keep
working with them.
13.9 - Calculus with Power Series
Let me recall a formula that we have seen for geometric series:
1
X
1
xn
MATH 09C 13.8 POWER SERIES (PART 2)
PAOLO MANTERO
Section 13.8: Power Series
Last time we have seen power series.
P
Theorem 1. For a given power series
cn xn there are only three possibilities:
P
cn xn is divergent for all x 6= 0
P
(and say that 0 is the
MATH 09C 13.8 POWER SERIES (PART 1)
PAOLO MANTERO
Recall: Last time we have seen the last Tests for Series (Root Test and Ratio
Test), that we now recall as we are going to need them a lot today.
Theorem 1 (The Ratio Test). Let an be a sequence.
P
n+1
I
MATH 09C - 13.6 ABSOLUTE CONVERGENCE AND
CONDITIONAL CONVERGENCE
PAOLO MANTERO
Last time: we have seen the Alternating Series Test and the Absolute Convergence Test.
Theorem 1 (Alternating Series Test). If a sequence an satises these three conditions:
(i
MATH 09C - 13.4 ALTERNATING SERIES AND ABSOLUTE
CONVERGENCE TEST
PAOLO MANTERO
Last time: we have seen the Limit Comparison Test and the Direct Comparison
Test.
Theorem 1 (Limit Comparison Test). Assume both an 0 and bn 0. If
an
lim
=c
n!1 bn
P
P
where c
MATH 09C - 13.3 - INTEGRAL TEST, p-SERIES TEST AND
ESTIMATES OF REMAINDERS
PAOLO MANTERO
Last time: we have seen
Telescopic Series.
P1
Test for Geometric Series: n=N kxn is convergent if |x| < 1 and divergent
otherwise.
P
Test for Divergence: if an 6!
9C - 13.1 SEQUENCES - PART 3: MORE ON LIMITS OF
SEQUENCES
PAOLO MANTERO
Last time we have seen that a sequence is convergent if limn! an exists and is
nite. If the limit does not exist or is innite, the sequence is said to be divergent.
To check whether a
MATH 09C - 13.2 - 13.3 - TEST FOR DIVERGENCE,
TELESCOPIC SERIES AND INTEGRAL TEST
PAOLO MANTERO
Recall: Last time we have seen what is a series.
P1
A series is an innite sum. A series n=1 an is just
1
X
an = lim sn
n!1
n=1
where sn is the sequence of the
MATH 09C 13.11 TAYLORS THEOREM (PART 2)
PAOLO MANTERO
Recall:
Last time we have seen that some fundamental functions can be represented as
a power series as follows:
P1 n
et = n=0 t near t = 0;
P1 n!
1 1 t = n=0 tn near t = 0;
P1 xn+1
P1 xn
ln(1 t) =
MATH 09C - 13.1 - SEQUENCES - PART 5: MONOTONIC AND
BOUNDED SEQUENCES
PAOLO MANTERO
Recall: Last time we have seen a few special limits, in particular
1
limn!1 nr = 0 for any r > 0.
p
n
limn!1 x = 1 for all x > 0,
n
limn!1 x = 0 for any x > 0, and
n!
1
MATH 09C - 13.1 - SEQUENCES - PART 4: SPECIAL LIMITS
AND ALTERNATING SEQUENCES
PAOLO MANTERO
Recall: Last time to compute limit of sequences we have employed
De LHospital Rule;
Limit of sequences of powers an = nt and an = rn ; and
Continuous functions
University of California, Riverside
Department of Mathematics
Midterm
Mathematics 9C - First Year of Calculus
Sample 3
Instructions: This exam has a total of 60 points. You have 50 minutes.
You must show all your work to receive full credit You may use an
Name:
Student ID Number:
TA Name / Discussion Time:
University of California, Riverside Department of Mathematics
Midterm
Mathematics 09C - First Year Calculus - Fall 2013
Instructions: This exam has a total of 60 points.
You have 50 minutes.
To receive
9C - ANSWERS OF FINAL SAMPLES - SPRING 2013
PAOLO MANTERO
Midterm Sample 1
(1) C to 0 (use De LHospitals Rule)
(2) This series is telescopic, so one can check that
(a) sn = 1 21
n
2
(b) the series is convergent and its sum is limn!1 sn = 1 .
2
(3) Conditi
University of California, Riverside
Department of Mathematics
Final Exam
Mathematics 9C - First Year of Calculus
Sample 3
Instructions: This exam has a total of 100 points. You have 3 hours.
You must show all your work to receive full credit. You may use
Section 9.3 Polar Coordinates
2010 Kiryl Tsishchanka
Polar Coordinates
DEFINITION: The polar coordinate system is a two-dimensional coordinate system in which each
point P on a plane is determined by a distance r from a xed point O that is called the pole
9C - 12.1 - POLAR COORDINATES - FALL 2013
PAOLO MANTERO
Section 12.1: Polar Coordinates
The very last page of these notes contains nice pictures of a few polar graphs.
Check them out! (you will probably need to zoom-in a little bit, since I tried to
keep
MATH 09C - 12.3 AREAS IN POLAR COORDINATES
PAOLO MANTERO
In the last two classes we have seen that the usual rectangular coordinates can
be written in terms of the polar coordinate r and as follows:
x = r cos()
and y = r sin()
Last class, we have seen th
MATH 09C 12.4 PARAMETRIC EQUATIONS
PAOLO MANTERO
Recall: Last time we have seen how to compute areas in Polar Coordinates:
Formula for Area in Polar Coordinates: The area of the region bounded by
r = f () between = a and = b is
A=
1
2
b
r2 d =
a
1
2
b
f
MATH 09C 12.5 - CALCULUS WITH PARAMETRIC
EQUATIONS
PAOLO MANTERO
This is the last class for Math 09C. Thanks for a nice quarter together, and good
luck with your nals!
Recall: Last time we have seen curves dened by parametric equations
x = f (t)
and
y =
9C - ANSWERS OF FINAL SAMPLES - SPRING 2013
PAOLO MANTERO
Final Sample 2
n+1 n
n
Recall that lim
n!1
= lim 1 +
n!1
1 n
n
= e.
(1) (a) 1 (use De LHospitals Rule);
n
1
1
(b) 1 (because n+1 = n+1 = 1+ 1 so the limit becomes (1+11 )n ! 1 ).
e
e
n
n
n
1
P
1 n
University of California, Riverside
Department of Mathematics
Final Exam
Mathematics 9C - First Year of Calculus
Sample 2
Instructions: This exam has a total of 100 points. You have 3 hours.
You must show all your work to receive full credit. You may use
University of California, Riverside
Department of Mathematics
Final Exam
Mathematics 9C - First Year of Calculus
Sample 1
Instructions: This exam has a total of 100 points. You have 3 hours.
You must show all your work to receive full credit. You may use
University of California, Riverside
Department of Mathematics
Final Exam
Mathematics 9C - First Year of Calculus
Sample 3
Instructions: This exam has a total of 100 points. You have 3 hours.
You must show all your work to receive full credit. You may use