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SOLUTIONS TO MIDTERM #1: MATH 102, SECTIONS 102 & 105
QUESTION 1:
[6 marks]
(a) Give the deﬁnition of the derivative
f
0
(
x
)o
fafunct
ion
f
(
x
)
.
(b) In a few brief sentences give several interpretations of the derivative of a function
f
(
x
)
.
(c) State the formula for the tangent line approximation.
(a) The derivative is deﬁned to be
f
0
(
x
0
) = lim
h
→
0
f
(
x
0
+
h
)

f
(
x
0
)
h
.
(b) The derivative
f
0
(
x
0
) can be interpreted as the slope of the graph
y
=
f
(
x
)when
x
=
x
0
.
It is also the instantaneous rate of change of
y
=
f
(
x
) with respect to
x.
If
y
=
f
(
t
)represents
distance travelled than
y
0
(
t
)=
v
(
t
) represents velocity and
v
0
(
t
y
00
(
t
a
(
t
)rep
re
s
en
t
s
acceleration.
(c) The tangent line approximation is
f
(
a
+
h
)
≈
f
(
a
)+
hf
0
(
a
)
.
Equivalently the tangent line
approximation is
f
(
a
+
h
f
(
a
hf
0
(
a
E
(
h
)
,
where
E
(
h
) represents the error made in
the approximation
f
(
a
+
h
)
≈
f
(
a
hf
0
(
a
)
.
y=f(x)
a
a+h
E(h)
f(a)+hf’(a)
y=f(a)+(xa)f’(a)
1
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QUESTION 2:
[6 marks]
(a) Using only the deﬁnition of the derivative, and not the rules of diﬀerentiation, ﬁnd the
derivative of
f
(
x
)=1
/
√
x.
(b) Find the equation of the tangent line of
y
=1
/
√
x
when
x
=25
.
(c) Using the tangent line approximation estimate the value of 1
/
√
26
.
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 Winter '06
 denissjerve

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