MAT1332A - Calculus for the Life Sciences II - Fall 2009
Assignment 3 This assignment is worth a total of 50 points. Due on Thursday, November 19, at the beginning of the lecture. Sorry, no late assignments will be accepted. Please write your answers neat
MAT 1332. Winter 2016, Assignment 6
Due Monday April 11 in the math department dropboxes by 7:00pm.
Late assignments will not be accepted; nor will unstapled assignments.
Professors in the math department will not lend you a stapler; do not ask for one.
I
MAT 1322B,
Winter 2005
Professor: Venanzio Capretta
MIDTERM TEST 3, version A Solutions
1. [2 points, 8.6 #8] Find a power series representation of the function: f (x) =
3 n 2n n=0 ( 2 ) x n=0
3x 1 2x2 6n xn+1 C. F.
2 2n+2 n=0 3n x n=0
A. D.
B. E.
n=0
MAT 1322E,
Winter 2005
MIDTERM TEST 2Solutions
Family NAME: Given NAME: Student Number: MAX=18 Time: 80 min. (Includes time for distribution of papers.) Only TI 30-type calculators are permitted. Notes or books are not permitted. There are 7 problems. Pro
MAT 1322, Winter 2005
SOLUTIONS TO TEST 1 (Version 1)
1. [2 points, 5.10 #17] Determine if the integral is convergent. A. 0.25 B. 0.11 C. 0.125 D. 0.333 Solution. By parts: u = x, v = e , u = 1, v = 1 1 3x 3x 3x dx = 3 xe +3 e dx 0 0 xe . = 1 xe3x 1 e3x 0
MAT 1322 MIDTERM 1, VERSION A
1 1. [2 points, 5.10 #19] Determine if the integral 2 x(ln x)2 dx is convergent or divergent and evaluate if it is convergent. A.1 B.2.51 C.0.50 D.5.33 E.1.44 F.divergent
Solution.
1 dx 2 x(ln x)2 1 = u ln 2 =
= 0
1 du ln 2
MAT 1322, Winter 2005 SAMPLE EXAMINATION MAX = 46 points
This sample exam is last years nal exam slightly modied to t this yearss format. The correct answers of multiple choice problems are indicated by bold-type.
Time: 3 hours Only TI 30-type calculators
MAT 1322 Winter 2004
MIDTERM TEST 3 Version A
1. [2 points, 8.8 #7] Find the MacLaurin series c0 + c1 x + c2 x2 + c3 x3 + of 1 the function 9+6x . The coecient c3 and the radius of convergence R is A. c3 = 20/81, R = 9 C. c3 = 5/72, R = 3/2 E. c3 = 27/128
Mat 1322D, Calculus 2 Professor. Arian Novruzi Department of Mathematics email: novruzi@uottawa.ca tel: (613) 562 5800 x 3530 Lastname, Firstname: Student number:
November 3rd, 2004
MIDTERM 2 (time 80 min)
Notes 1) No books or any other document are allow
Tutorial #10 - MAT 1332A - Fall 2009 November 23, 2009
Find the eigenvalues and eigenvectors of A = Solution: 1 = 2 + 2i, v1 = 1 2 + 2i
0 8
1 4
. 1 2 2i
; 2 = 2 2i, v2 =
On any given day, a student is either healthy or ill. Of the students who are healt
MAT 1322,
MIDTERM 1,
VERSION A
1. [2 points, 7.2 #19] Use Eulers method with step size 0.2 to estimate y(0.4), where y is the solution of the initial value problem y= 2x+y , y(0) =1. A. 1.2 B. 1.46 C. 1.56 D. 1.48 E. 1.52 F.1.44
Solution: We are given h =
University of Ottawa MAT 1332B Midterm Exam
Feb. 13, 2008. Duration: 80 minutes. Instructor: Frithjof Lutscher
Solutions
Question 1.
[3 points] Consider the two functions f (x) = 1 , 1 + 2x g(x) = 1 - x.
1. (1 point) Show that the functions intersect at t
MAT 1332, Winter 2015, Assignment 1
Due Thursday January 29 by 9:00pm.
Late assignments will not be accepted; nor will unstapled assignments.
Professors in the math department will not lend you a stapler; do not ask for one.
Instructor (circle one): Rober
April 19, 2012 Midterm MAT1332D
Winter 2012
Study guide - Final Exam
As you read these let me know if you think I have missed anything. I will add more as I
can.
The best way to study is:
1. make sure you can do all of the problems of the homework assignm
Professor Jason Levy, University of Ottawa, MAT 13320, Winter 2011
Assignment 3 Solutions
1. For each of the following improper integrals, determine whether it converges, and
evaluate it if if does.
(a) [0 t2 dt (b) /1°°e~tsin(2t)dt
4+2:6
Solution: a) W
MAT1332, Winter 2011, Assignment 1 Solutions
Total=7 points.
1. (1 point) (a) Use the substitution u = 3 1415. Then élt = 14, so dt = :114. Thus
/314tdt_ Ed_lnlul+0lnl314tl+0
(Dont forget to resubstitute - and dont forget the absolute value signs in the
MAT 1332, Winter 2009, Assignment #3, Solutions
Total marks=29. [2] 1. Using the substitution u = 2+5x, then du = 5dx, and then we find that dx = are from u = 2 to u = . Therefore,
0 du 5 ,
and the limits of integration
1 dx = (2 + 5x)4
2
1 -1 -3 1 -4 u
MAT 1332, Winter 2007, Assignment #2, Solutions
Total marks=11. [2] 1. If you simply integrate, the answer is zero. So that obviously isn't right. It's best to sketch the two graphs. See Figure 1.
1 A sin(4x) cos(4x) 0.5 B 0
C
0.5
1 /2 7 /16 3 /16 0
Figur
University of Ottawa MAT 1332B Midterm Exam
March 31, 2008. Duration: 80 minutes. Instructor: Frithjof Lutscher
Family Name:
First Name:
Do not write you student ID number on this front page. Please write your student ID number in the space provided on th
Tutorial #9 - MAT 1332A - Fall 2009 November 16, 2009
4 Let A = 2 0
0 5 0
2 4 . Find the eigenvalues and eigenvectors of A. 5
= Answer: 1 5 is an eigenvalue whose corresponding eigenvectors are 2 0 v1 = 0 and v2 = 1 . The other eigenvalue is 2 = 4 with 1
Tutorial #8 - MAT 1332A - Fall 2009 November 9, 2009
Section 5.1: questions 17 and 21 30 4 1 Let A = 1 2 , B = 0 2 11 following (where possible): (a) 2AT + C (b) AB (c) CB (d) tr(CC T ) (e) det A (f) det B Consider the following system:
,C=
1 3
4 1
2 5
Solutions to Assignment 2 - MAT1332A - Fall 2009 (1) (a) [6 points] Check that N (t) = equation t is a solution of the dierential 1 + ct
dN N2 = 2 . Treat c as an unspecied constant. dt t
(b) [5 points] Use N (1) = 1 to nd c. Then give the solution N (t)
Solutions to Assignment 1 - MAT1332A - Fall 2009
(1) Evaluate
/2
3 cos 2 d.
Solution: First, lets make a change of variable. Let u = 2 . So du = 2d. Then
1 cos d = 2
3 2 /2 /2
u cos u du.
Now we use integration by parts (let f (u) = cos u and g (u)
Systems of dierential equations (the complex case) and double integrals over rectangles 1. Find the general solution of the system: (a) x1 = 2x1 5x2 x2 = 4x1 2x2 x1 = x1 2x2 x2 = 2x1 + x2 x1 = 5x1 9x2 x2 = 2x1 x2 f (x, y ) dxdy :
R
(b)
(c)
2. Evaluate
(a)
The chain rule (general version) Find z/u and z/v by using the chain rule. (a) z = xey + yex , x = u sin v , y = v cos u (b) z = xey , x = ln u, y = v (c) z = xey , x = u2 + v 2 , y = u2 v 2 (d) z = sin(x/y ), x = ln u, y = v Answers: (a) z = (ev cos u v
Critical points and their classication, chain rule (the simplest version) 1. Find all critical points and classify them as local max, local min, or saddle point. (a) f (x, y ) = 2x2 + y 2 + 4x 4y + 5 (b) f (x, y ) = 6xy 2 2x3 3y 4 (c) f (x, y ) = x3 + y 3
Gradient and directional derivative 1. Find the grad f and the grad f (P ). (a) f (x, y ) = 3x 7y , P (17, 39) (b) f (x, y ) = 3x2 5y 2 , P (2, 3) (c) f (x, y ) = ex
2
y 2
, P (0, 0)
2. Find the directional derivative of f at P in the direction of . v (a
Partial derivatives and tangent plane Find an equation of the tangent plane to the given surface at the specied point. (1) f (x, y ) = x3 + 2xy y + 3, P0 = (1, 2, 6) (2) f (x, y ) = ey + x + x2 + 6, P0 = (1, 0, 9) (3) f (x, y ) = x y , P0 = (5, 1, 2) (4)