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MAT133Y5Y April 2012 Final examination-A
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UNIVERSITY OF TORONTO MISSISSAUGA
APRIL 2012 FINAL EXAMINATION Version A
MAT133Y5Y
Calculus and Linear Algebra for
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MAT133Y5Y April 2011 Final examination-A
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MAT133Y5Y
Calculus and Linear Algebra for Commerce
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MAT133Y5Y April 2015 Final Examination-A
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Calculus and Linear Algebra fo
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 PCBs in the Arctic lakes.
In order to conduct this study,
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 both sides with respect to x.
dy
4. Solve for
.
dx
5. Ex
