Math 117: Honours Calculus I
Fall, 2013
Assignment 1
September 6, due September 18
1. Prove that
3 is irrational.
Suppose that there existed integers p and q such that p2 = 3q 2 . Without loss of
generality we may assume that p and q are are not both divi
Math 117: Honours Calculus I
Dr. J. Bowman 13:00-13:50 November 17, 2005
Midterm Exam 2
Last Name:
Student ID:
First Name:
Question 1 2 3 4 5 Total
Score
Maximum 3 2 2 3 2 12
No calculators or formula sheets. Check that you have 3 pages.
1. Determine whic
Math 117 Fall 2014 Midterm Exam 2
Oct. 24, 2014 10am - 10:50am. Total 20+2 Pts
NAME:
ID#:
There are ve questions.
Please write clearly and show enough work.
1
2
3
4
5
Total
1
2
Math 117 Fall 2014 Midterm Exam 2
Question 1. (5 pts) Prove by denition:
lim x
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #2
Dr. Michael Li
#1.8 (20 point) Prove
(a)
n
0
+
n
1
+
n
2
+
n
n1
(b)
n
0
n
1
+
n
2
+ (1)n1
+
n
n
n
n1
= 2n .
+ (1)n
n
n
= 0.
Proof. (a). Set x = 1 in the following form of the binomial th
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #6
Dr. Michael Li
#2.14 (80 point) (a) Give a denition of
lim an = .
n
(b) Suppose limt an = , limt bn = . Use your denition to prove
lim (an + bn ) = ,
t
lim an bn = .
t
(c) Show by example
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #5
Dr. Michael Li
#2.8 (20 point) Suppose the sequence cfw_an is dened inductively by
a1 = 0,
an+1 =
3 + 2an ,
n = 1, 2, , 3, .
(1)
Show that limn an = 3.
Proof. After observing the rst few
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #4
Dr. Michael Li
#2.2 (20 point) (Squeeze Principle). Suppose xn zn yn , n = 1, 2,
and cfw_xn , cfw_yn are both convergent with limit c. Show that cfw_zn is also
convergent with limit c.
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #3
Dr. Michael Li
#1.10 (20 point) Check that
1
1
1
= ,
2
2
1
1
2
1
1
1
= ,
3
3
1
1
2
1
1
3
1
1
1
= .
4
4
Guess a general formula and prove by induction.
Solution. The general formula is
1
1
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #8
Dr. Michael Li
#3.15 (20 point) For each of the functions in #3.1, page 52, nd all points
where the function is not continuous. Explain briey in each case.
Solution.
(a) f (x) = 3x + 2 is
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #10
Dr. Michael Li
#5.1 (20 point) Let f (x) = |x|, 1 x 1. Then f (1) = f (1) = 1,
but there is no c (1, 1) for which f (c) = 0. Does this contradict Rolles
Theorem?
Solution. The answer is
Math 117 Section A1 Midterm Exam Fall 2003-04
Time: 50 Minutes
Instructor: Dr. Michael Li
Note:Show all your work on separate sheets of paper. Your answers worth 1/4 of the credit. Your
analysis and steps leading to the answer worth 3/4 of the credit.
1.
Math 117 Section A1 Midterm Exam II Fall 2003-04
Instructor: Dr. Michael Li
Solutions
1. (10%) State the Following Theorems.
1. Balzano-Weierstrass Theorem
2. Denition of Cauchy sequence and the Cauchy criteria for convergent sequences.
3. Intermediate Va
Math 117 Honors Calculus Fall 2003-2004
Solutions to Quiz #1
Dr. Michael Li
#1. (5 point) Prove
1
12 + 22 + 32 + + n2 = n(n + 1)(2n + 1), for all n N.
6
Proof 1. Use mathematical induction on n.
1. When n = 1, RHS= 1 1(1 + 1)(2 + 1) = 1 =LHS.
6
2. Assume
Math 117 Honors Calculus I Fall, 2003-2004
Solutions to Homework #11
Dr. Michael Li
#5.62 (20 point) Let f, be dierential functions such that
f g g f > 0.
(1)
Show that between any pair of roots of the equation f (x) = 0 there is a root
of g(x) = 0.
Proof
Math 117 Fall 2014 Homework 5 Solutions
Due Thursday Oct. 16 3pm in Assignment Box
Question 1. (5 pts) Let f ; g: R 7! R and a 2 R. Further assume limx!af (x) = L 2 R and
limx!ag(x) = M 2 R.
a) (2 pts) Prove or disprove: Under the above assumptions, there
Math 117 Fall 2014 Homework 6 Solutions
Due Thursday Oct. 30 3pm in Assignment Box
Question 1. (5 pts) Let
P1
n=1
1
X
an be an innite series. Prove that
an converges =)
n=1
Proof. Let " > 0 be arbitrary. As
for every m > n > N1,
P1
n=1
lim an = 0:
n!1
(1)
Math 117: Honours Calculus I
Dr. J. Bowman 13:00-13:50 November 9, 2006
Midterm Exam 2
Last Name:
Student ID:
First Name:
Question 1 2 3 4 5 6 Total
Score
Maximum 2 3 3 2 2 2 14
No calculators or formula sheets. Check that you have 2 pages.
1. Determine w
Math 117: Honours Calculus I
Dr. J. Bowman 13:00-13:50 October 20, 2005
Midterm Exam 1
Last Name:
Student ID:
First Name:
Question 1 2 3 4 Total
Score
Maximum 5 3 3 7 18
No calculators or formula sheets. Check that you have 2 pages.
1. Determine which of
Math 117: Honours Calculus I
Dr. J. Bowman 13:00-13:50 October 11, 2012
Midterm Exam 1
First Name:
Last Name:
Question 1 2 3 Total
Score
Maximum 3 8 5 16
Student ID:
No calculators or formula sheets. Check that you have 2 pages.
1. Determine which of the
Math 117: Honours Calculus I
Fall, 2013
Assignment 4
October 10 due October 21
1. Extend the result
0 if 0 c < 1,
n
lim c = 1 if c = 1,
n
/ if c > 1
to show that (here n takes on only integer values)
0 if |r| < 1,
n
lim r = 1 if r = 1,
n
/ if r 1 or r >
Math 117: Honours Calculus I
Fall, 2013
Assignment 7
November 14, due November 25
1. (a) A spherical balloon is being inflated at the rate of 10 cm3 /s. Given that the
volume V of the balloon is related to the radius by V = 34 r 3 , use the Chain
Rule to
Math 117: Honours Calculus I
Fall, 2013
Assignment 5
October 22 due 16:00 October 30 (no extensions)
1. Let cfw_an
n=1 be a sequence of real numbers. Suppose that the two subsequences
cfw_a2k k=1 and cfw_a2k+1
k=1 converge to the same limit L.
(a) Prove
Math 117: Honours Calculus I
Dr. J. Bowman 13:00-13:50 October 19, 2006
Midterm Exam 1
Last Name:
Student ID:
First Name:
Question 1 2 3 Total
Score
Maximum 4 4 6 14
No calculators or formula sheets. Check that you have 2 pages.
1. Determine which of the
Math 117: Honours Calculus I
Fall, 2013
Assignment 4
October 10 due October 18
1. Extend the result
0 if 0 c < 1,
n
lim c = 1 if c = 1,
n
/ if c > 1
to show that (here n takes on only integer values)
0 if |r| < 1,
n
lim r = 1 if r = 1,
n
/ if r 1 or r >
Math 117: Honours Calculus I
Fall, 2013
Assignment 5
October 22 due 16:00 October 30 (no extensions)
1. Let cfw_an
n=1 be a sequence of real numbers. Suppose that the two subsequences
cfw_a2k
k=1 and cfw_a2k+1 k=1 converge to the same limit L.
(a) Prove
Math 117: Honours Calculus I
Fall, 2008
Assignment 8
November 22, due December 4
1. Prove that the sum of a positive number and its reciprocal is at least 2, using
(a) the First Derivative Test.
(b) the Second Derivative Test. Use Corollary 4.5.3 to show
Math 117: Honours Calculus I
Fall, 2013
Assignment 3
September 30 due October 8
1. Determine which of the following limits exist, using either the properties of
limits or directly from the , N definition (your choice). For those that exist,
compute the li
Math 117: Honours Calculus I
Fall, 2013
Assignment 6
November 5 due November 14
1. (a) Show that f (x) = x7 + x5 + 2x 1 has at least one real root in (0, 1).
(b) Prove that f (x) = sin5 x sin3 x + sin2 x
(0, ).
1
2
has at least one real root in
2. Let T
Math 117: Honours Calculus I
Fall, 2013
Assignment 7
November 14, due November 22
1. (a) A spherical balloon is being inflated at the rate of 10 cm3 /s. Given that the
volume V of the balloon is related to the radius by V = 34 r 3 , use the Chain
Rule to