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MAT133Y5Y April 2012 Final examination-A
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APRIL 2012 FINAL EXAMINATI
[email protected] (20162017)
Calculus and Linear Algebra for Commerce
Problem Set M to be submitted to your TA
Due Date:
This assignment must be submitted to YOUR TA at the beginning of YOUR tutorial,
MCS @UTM
MAT 133Y Term Test #3 Version XXX
January 31, 2014, 4pm
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th
I'The'natUral exponential and logarithmiC'fUnctiOns .
(inverse functions), with many important properties:
-y = loga'x => x = ay
Note that _ _
y=logx=logmx x=10
y=lnx=lOgex'<:>x=ey
e"e = e"
a-b
Fact Sheet on APPLICATIONS of INTEGRATION
Finding the area between curves.
- Need a rough sketch
- Need to know the points of
intersection
- Write and calculate the appropriate
definite integral
For e
Fact Sheet #3 (on Matrix Algebra)
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Fact Sheet #3 (on Matrix Algebra)
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Some Properties of Matrices:
Assuming the sizes of the matrices are such that the operations may be performe
Fact Sheet #4
Page 1 of 4
Given that lim f ( x) = L and lim g( x) = M both
xa
xa
exist and c is a constant:
1. lim c = c
xa
2. lim x n = a n
xa
[limit of a constant]
[limit of a power of x]
3. lim f (
The Derivative and its Applications
Fact Sheet (single variable)
Limit definition of the derivative:
f ( x ) = lim
h0
f ( x + h) f ( x )
h
Some differentiation rules/techniques:
1.
d
[c ] = 0
[derivat
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MAT133Y5Y April 2011 Final examination-A
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UNIVERSITY OF TORONTO MISSISSAUGA
APRIL 2011 FINAL EXAMINATION-Version A
MAT
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MAT133Y5Y April 2015 Final Examination-A
Student number_
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UNIVERSITY OF TORONTO MISSISSAUGA
APRIL 2015 FINAL
An increase to precipitation (rain/snow) in the Arctic will lead to higher levels of PCBs due to
atmospheric circulation.
Precipitation due to atmospheric circulation has no effect on the levels of PC
MAT133Y5Y @UTM - Calculus and Linear Algebra for Commerce
COURSE INFORMATION and OUTLINE - 2016-2017
Description
MAT133Y5Y is an introductory survey of some basic theory and applications of Calculus a
PARTIII:CALCULUS
Ch.17:MultivariableCalculus
17.1PartialDerivatives
Definition
Ifz=(x,y),thepartialderivativeoffwithrespecttox,denotedfx,isthefunction,oftwo
variables,givenby
f ( x +h , y )f ( x , y )
MGM101 Lecture
Nov 23th, 2016/11/23
1. In the news
Investment about IPO plan
Long road, camera ready and strong interest
Competitive priority
emphasis
examples of organization
Cost
low cost
Wal-mart
2
MAT133-MAR-2015Term Test 4A
March 27, 2015, 4pm
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Page 2 of 13
For each question 1 to 10, you are required to show all your work. A correct answer obtained with false reasoning
MCS @UTM
MAT 133Y Term Test #1 Version A
October 9, 2015, 5pm
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MCS @UTM
MAT 133Y Term Test #4A
March 18, 2016, 5pm
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the appropriate
MCS @UTM
MAT 133Y Term Test #1 Version A
October 10, 2014, 4pm
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NAME (PRINT): -.:~=-=-_ _ _ _ _ _~:-:-;-;-:_ _ _ _ _ _ _
LasUSumame
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UNIVERSITY OF TORONTO MISSISSAUGA
APRIL 2012 FINAL E
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APRIL 2013 FINAL EXAMINATION-Version B
MAT133Y5Y
Calculus and Linear Algebra for Comme
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UNIVERSITY OF TORONTO MISSISSAUGA
APRIL 2016 FINAL EXAMINATION-A
MAT133Y5Y
Calculus and Linear Algebra for Commerce
Alex
12.5 Logarithmic Differentiation
To differentiate y=f (x) ,
1. Take the natural logarithm of both sides.
lny=ln ( f ( x )
2. Simplify ln ( f ( x ) by using properties of logarithms.
3. Differentiate b