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MATHEMATICS 105 (Section 921) - Quiz #1
SOLUTIONS
INSTRUCTIONS: Notes, books and calculators are not permitted. Answers may receive no credit if accompanying work is not provided. When indicated, simplify answers as much as possible, and be sure to clearly indicate the final answer for each question. Please use the back of the pages if you need extra space.
5 1.
For the function f ( x , y ) = ln ( x − 3y ) , use the tangent plane approximation at ( x , y ) = ( a , b ) = ( 7 , 2 ) to estimate the value of f ( 6 .
9 , 2 .
06 ) .
The tangent plane/linear approximation formula is: f ( x , y ) ≈ f ( a , b ) + f x
( a , b ) · ( x − a ) + f y
( a , b ) · ( y − b ) where here we have x = 6 .
9 , y = 2 .
06 , a = 7 , b = 2. We can also calculate f ( a , b ) = f ( 7 , 2 ) = ln ( 1 ) = 0. We need the partial derivatives f x
=
1 x − 3y
· 1 , f y
=
1 x − 3y
· ( − 3 ) so we have
1 f x
( 7 , 2 ) =
7 − 6
= 1 , f y
=
− 3
7 − 6
= − 3
Plugging everything into the approximation formula, we now have f ( 6 .
9 , 2 .
06 ) ≈ 0 + 1 · ( 6 .
9 − 7 ) + ( − 3 ) · ( 2 .
06 − 2 )
= − 0 .
28
So we have the approximation f ( 6 .
9 , 2 .
06 ) ≈ − 0 .
28.

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5 2.
Find all the critical points of the function f ( x , y ) = x
3
− 6xy + 3y
2
− 9x + 7 .
Critical points are where both partial derivatives are zero or at least one is undefined. Here we have f x
= 3x
2
− 6y − 9 , f y
= − 6x + 6y both of which are defined everywhere. So, we must solve the system,
3x
2
− 6y − 9 = 0
− 6x + 6y = 0
From the second equation we have y = x. Substituting this in for y in the first equation we have
3y
2
− 6y − 9 = 0 factoring this gives
3 ( y − 3 )( y + 1 ) = 0 so that y = 3 or y = − 1. Then, since x = y we have that the critical points are:
( 3 , 3 ) and ( − 1 , − 1 )