It follows from
that the function
f
defined by
lim
x
→
0
sin(
x
)
x
1.
is continuous at
x
= 0.
Since sine is continuous and onetoone on
[
π
/2,
π
/2], sin
1
is onetoone and continuous on [1,1], and thus,
lim
x
→
0
sin
1
(
x
)
sin
1
(0)
0.
Consequently, applying Theorem 2.5.5 with
f
as above,
g
(x) = sin
1
(
x
) and
c
= 0, since sin
1
(
x
)
≠
0 when
x
≠
0, we have
lim
x
→
0
x
sin
1
(
x
)
lim
x
→
0
sin(sin
1
(
x
))
sin
1
(
x
)
lim
x
→
0
f
(
g
(
x
))
f
(0)
1.
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______________________________________________________________________
5. (10 pts.)
What are the x and y  intercepts of the tangent line to
the graph of y = 1/x
2
at the point (2,1/4)?
The slope of the tangent line at x = 2 is
dy
dx
x
2
2
x
3
x
2
1
4
.
Consequently, with a little routine algebra, we see that an equation for
the tangent line in slopeintercept form is given by
y
1
4
x
3
4
.
The y intercept is plainly
y
= 3/4 and one can see with a little work that
the x intercept is
x
= 3.//
______________________________________________________________________
6. (10 pts.)
(a)
Find all values in the interval [0,2
π
] at which the
graph of
f
has a horizontal tangent line when
f
(
x
) =
x
+ 2 cos(
x
).
Since
f
has a horizontal tangent line when
f
′
(
x
) = 0, and
f
′
(
x
) = 1  2sin(
x
), it follows that
f
has a horizontal tangent line in the
interval [0,2
π
] when sin(
x
) = 1/2.
This happens only at
x
1
=
π
/6 or
x
2
= 5
π
/6 .
(b)
The following limit represents
f
′
(
a
) for some function
f
and some
number
a
.
Using that information, evaluate the limit.
lim
x
→ π
sin(3
x
)
0
x
π
3.
Here, of course,
f
(
x
) = sin(3
x
) and
a
=
π
. So we have
lim
x
→ π
sin(3
x
)
0
x
π
f
(
π
)
3cos(3
π
)
3(
1)
3
since f
(
x
)
3cos(3
x
).
______________________________________________________________________
7. (10 pts.) (a)
Using complete sentences and appropriate notation,
provide the precise mathematical definition for the derivative,
f
′
(
x
), of a
function
f
(
x
).//
The function
f
′
defined by the equation
f
(
x
)
lim
h
→
0
f
(
x
h
)
f
(
x
)
h
.
is called the derivative of
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 Fall '08
 STAFF
 Calculus, Derivative, Continuous function, lim g

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