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MATH 408N PRACTICE MIDTERM 2
Name:
10/21/2011
Bormashenko
TA session:
Show your work for all the problems. Good luck!
(1) Use the limit deﬁnition of the derivative for the following questions.
You will get no points for
using the rules learned later!
(a) [5 pts] Find
f
0
(1) if
f
(
x
) =
√
x
.
(b) [5 pts] Find
f
0
(
x
) if
f
(
x
) =
x
2
+
x
+ 1.
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MATH 408N PRACTICE MIDTERM 2
(2) Diﬀerentiate the following functions, using whatever methods you think are best (you are now
allowed to use all the rules):
(a) [5 pts]
f
(
x
) =
x
3
+ 3
x
+ 5
(b) [5 pts]
f
(
x
) = arctan
(
x
2
)
e
x
MATH 408N PRACTICE MIDTERM 2
3
(c) [5 pts]
f
(
x
) =
2
x
sin(
x
)
ln(
x
+ 1)
(d) [5 pts]
f
(
x
) = sin(
x
)
cos(
x
)
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MATH 408N PRACTICE MIDTERM 2
(3) Find the equations of the tangent lines to the following graphs at the given points:
(a) [7 pts]
y
=
x
2
ln(
x
) + 1 at (1
,
1).
(b) [8 pts]
x
2
+
y
2
=
xe
y
at (1
,
0).
MATH 408N PRACTICE MIDTERM 2
5
(4) Let
f
(
x
) and
g
(
x
) satisfy
f
(1) = 3
,g
(1) = 2
,f
0
(1) = 1
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This note was uploaded on 11/06/2011 for the course MATH  taught by Professor  during the Spring '11 term at University of Texas at Austin.
 Spring '11
 
 Derivative

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