L IST OF SUGGESTED EXERCISES FOR CALCULUS 1501B - 2009-2010 PROVING INEQUALITIES: APPLICATIONS OF THE M EAN VALUE THEOREM Chapter 4 WWW Additional exercises will be provided T ECHNIQUES OF INTEGRATION SECTION 7 .1 7 .2 7.3 7.4 7.5 7.8 Ch. 7 Review PAGE 45
NAME:.
STUDENT NUMBER:.
THE UNIVERSITY OF WESTERN ONTARIO (WESTERN UNIVERSITY)
CALCULUS 1501B, Winter 2016
Midterm examination 1
Friday, February 5, 2016, 7:00-9:00 pm
INSTRUCTIONS:
SHOW ALL OF YOUR WORK AND PROVIDE CLEAR EXPLANATION. All results
must be
NAME:.
STUDENT NUMBER:.
THE UNIVERSITY OF WESTERN ONTARIO (WESTERN UNIVERSITY)
CALCULUS 1501B, Winter 2016
Midterm examination 2
Friday, March 11, 2016, 7:00-9:00 pm
INSTRUCTIONS:
SHOW ALL OF YOUR WORK AND PROVIDE CLEAR EXPLANATION. All results
must be ju
Tuesday, April 29, 2014
Page 1
Calculus 1501B
Final Examination
5
1. Suppose that f (1) = 2, f (4) = 7, f (1) = 5, f (4) = 3, and f is continuous. Find the
marks
value of
Z 4
xf (x) dx
1
Tuesday, April 29, 2014
Page 2
Calculus 1501B
Final Examination
10
2
Tuesday, April 29, 2014
Page 1
Calculus 1501B
Final Examination
5
1. Suppose that f (1) = 2, f (4) = 7, f (1) = 5, f (4) = 3, and f is continuous. Find the
marks
value of
4
xf (x) dx
1
Tuesday, April 29, 2014
Page 2
Calculus 1501B
Final Examination
10
2.
Calculus 1501B
Midterm Examination 2
5
1.
marks
Friday, March 14, 2014
Page 1
(a) (3) + (4) =
(b) If x0 > 1 is such that (x0 ) = 3 , then
2
(x0 + 1)
=
2x0
Calculus 1501B
Midterm Examination 2
Friday, March 14, 2014
Page 2
4
2. Prove that if limn an = 0 an
Written Assignment #1 - SOLUTIONS
Question 1. Use the denitions of continuity and dierentiability to prove that the function
f (x) =
x cos(1/x)
0
if x = 0
if x = 0
is continuous at 0 but not dierentiable at 0.
We begin by proving that f (x) is continuous
Written Assignment #2 - SOLUTIONS
Question 1. Evaluate the following integrals
1.
2.
ln3 (x)dx
ln2 (x + x + 12 )dx
1. We proceed using Integration by Parts with
u = ln3 (x)
du =
dv = dx
3 ln2 (x)
x
v=x
ln3 (x)dx = x ln3 (x) 3
ln2 (x)dx .
We apply Integrat
Calculus 1501B
Winter 2015
Written Assignment 4
due: Mar. 26 (Sec.002) / Mar. 27 (Sec.001), in class
1
1. (a) Let f (x) = e x2 , x > 0. Use mathematical induction to prove that, for every n 1, the
1
nth derivative f (n) (x) is of the form Pn (1/x) e x2 fo
i<
lt-l.-l,$1
yw
I " lz'-:.l Py+ f+8 ,e 4n'.l,dv sI ry'o<!
| - cv2
h *fi
%'Z orcetf
t *rf+ '.*.aflwrrttt |av o+44 rJ xaoy)
f'7*ry
Z < [ cw]
s.r
- ,;r]l 'g z lo -xl ?t
.
;75
(:<f aprd
'o
1gA ,?s 'a.l
t
Y#
w y ya,s' ! r*,i utn,i :' .';
"
f rry>xzxv1s7 o=e,f
Calculus 150113 Friday, Februaryr 7, 2014
Midterm Examination 1 Page 1
15’ 1.
marks {4 marks) (a) State the Mean Value Theorem precisely.
u etch.) 1E0) .s C9u4u‘n.mu£ on {[2,5]}
w¥W€.s*\lﬂL’e€ on (1,5)
hum 3ce(q,b) g_-1_ {Tchﬂi—fra)
*O.
(4' marks) (b) D
Calculus 1501B Winter 2011
Midterm 2 Solutions
Midterm 2 Solutions
1. Determine the radius and interval of convergence for the power series
n
(x + 3)
.
2n n
n=1
Solution:
I. Let an =
(x+3)n
,
2n n
and assume x + 3 = 0. Then
an+1
|x + 3|n+1
2n n
|x + 3|
THE UNIVERSITY OF WESTERN ONTARIO
LONDON
CANADA
DEPARTMENT OF MATHEMATICS
Calculus 1501B Midterm Examination 1
Friday, February 6, 2015
7:00 9:00 p.m.
1. (a) State the denition for the limit of a function at a point.
(b) Use this denition to prove that
li
Calculus 1501B
Midterm Examination 2
5
1.
marks
Friday, March 14, 2014
Page 1
(a) (3) + (4) =
(b) If x0 > 1 is such that (x0 ) = 32 , then
(x0 + 1)
=
2x0
Calculus 1501B
Midterm Examination 2
Friday, March 14, 2014
Page 2
4
2. Prove that if limn an = 0 and
Calculus 1501B
Midterm Examination 1
13
1.
marks
Friday, February 7, 2014
Page 1
(4 marks) (a) State the Mean Value Theorem precisely.
(4 marks) (b) Does there exist a function f which is differentiable everywhere and
f (1) = 4, f (4) = 10, f (x) 1 for al
Written Assignment #4 - SOLUTIONS
Question 1.
1
1. Let f (x) = e x2 , x > 0. Use mathematical induction to prove that, for every n 1, the
1
n-th derivative f (n) (x) is of the form Pn (1/x) e x2 for some polynomial Pn (depending
on n).
2. Dene
g(x) =
if x
Calculus 1501B
Winter 2015
Written Assignment 4
due: Mar. 26 (Sec.002) / Mar. 27 (Sec.001), in class
1
1. (a) Let f (x) = e x2 , x > 0. Use mathematical induction to prove that, for every n 1, the
1
nth derivative f (n) (x) is of the form Pn (1/x) e x2 fo
Written Assignment #3 - SOLUTIONS
Question 1. Let cfw_an and cfw_bn be convergent sequences. Use the N denition of
n=1
n=1
limit to prove that the product sequence cfw_an bn is also convergent and that
n=1
lim (an bn ) = lim an lim bn
n
n
n
If cfw_an
Calculus 1501B
Winter 2015
Written Assignment 3
due: Mar. 5 (Sec.002) / Mar. 6 (Sec.001), in class
1. Let cfw_an and cfw_bn be convergent sequences. Use the
n=1
n=1
that the product sequence cfw_an bn is also convergent and
n=1
N denition of limit to
Calculus 1501B
Winter 2015
Written Assignment 2
due: Feb. 12 (Sec.002) / Feb. 13 (Sec.001), in class
1. Evaluate the following integrals:
(a)
ln3 x dx
(b)
ln2 (x +
1 + x2 ) dx.
2. Determine for which values of , R the following improper integrals converge
Written Assignment #2 - SOLUTIONS
Question 1. Evaluate the following integrals
1.
2.
ln3 (x)dx
ln2 (x + x + 12 )dx
1. We proceed using Integration by Parts with
u = ln3 (x)
du =
dv = dx
3 ln2 (x)
x
v=x
ln3 (x)dx = x ln3 (x) 3
ln2 (x)dx .
We apply Integrat
Written Assignment #1 - SOLUTIONS
Question 1. Use the denitions of continuity and dierentiability to prove that the function
f (x) =
x cos(1/x)
0
if x = 0
if x = 0
is continuous at 0 but not dierentiable at 0.
We begin by proving that f (x) is continuous
Calculus 1501B
Winter 2015
Written Assignment 1
due: Jan. 22 (Sec.002) / Jan. 23 (Sec.001), in class
1. Use the denitions of continuity and dierentiability to prove that the function
f (x) =
xcos(1/x)
0
if x = 0
if x = 0
is continuous at 0 but not dierent
Friday, March 15, 2013
Page 1
Calculus 1501
Second Midterm Examination
n
, is cfw_an monotonic? Is it bounded? Explain.
10
1. If an = (1)n
n=1
n+1
marks
Calculus 1501
Second Midterm Examination
Friday, March 15, 2013
Page 2
1
5
2. Let an = n . Use the fo
Calculus 1501B
Midterm Examination 1
8
1.
marks
Friday, February 7, 2014
Page 1
(a) State the Mean Value Theorem precisely.
(b) Does there exist a function f which is dierentiable everywhere and f (1) = 4,
f (4) = 10, f (x) 1 for all x between 1 and 4? Ex
Calculus 150113 Friday, Februaryr 7, 2014
Midterm Examination 1 Page 1
15 1.
marks cfw_4 marks) (a) State the Mean Value Theorem precisely.
u etch.) 1E0) .s C9u4un.mu on cfw_[2,5]
wW.s*\lLe on (1,5)
hum 3ce(q,b) g_-1_ cfw_Tchifra)
*O.
(4' marks) (b) Doe