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SOLUTIONS TO MIDTERM #1
QUESTION 1:
[10 marks]
(a) Give the deﬁnition of the derivative
f
0
(
x
)o
fafunct
ion
f
(
x
).
(b) What is the equation of the tangent line to the graph of
y
=
f
(
x
)at
x
=
a
?
(c)
Using only the deﬁnition of the derivative
ﬁnd
f
0
(
x
) for the function
f
(
x
)=
1
√
x
.
Solutions:
(a)
f
0
(
x
) = lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
.
(b)
y

f
(
a
f
0
(
a
)(
x

a
)
.
(c) The derivative of
f
(
x
√
x
is calculated as follows:
f
0
(
x
) = lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
= lim
h
→
0
1
/
√
x
+
h

1
/
√
x
h
= lim
h
→
0
√
x

√
x
+
h
h
√
x
√
x
+
h
= lim
h
→
0
±
√
x

√
x
+
h
h
√
x
√
x
+
h
×
√
x
+
√
x
+
h
√
x
+
√
x
+
h
²
= lim
h
→
0
x

(
x
+
h
)
h
√
x
√
x
+
h
(
√
x
+
√
x
+
h
)
= lim
h
→
0

1
√
x
√
x
+
h
(
√
x
+
√
x
+
h
)
=

1
2
x
√
x
by putting
h
= 0 and using continuity.
QUESTION 2:
[12 marks]
(a) Find the derivative of
f
(
x
p
g
(
x
x
=

1i
f
g
(

1) = 4 and
g
0
(

1) =

2.
(b) Find the derivative of
f
(
x
x

1
g
(
x
)+1
at
x
=0if
g
(0)=2and
g
0
(0) = 2.
(c) Find the derivative of
f
(
x
)=s
in
(
πg
(
x
)) at
x
=
a
if you are given that
g
(
a
2
3
and
g
0
(
a
b
.
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 Winter '06
 denissjerve

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