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WORKSHEET 7  Fall 1995
1. Let
f
(
x
)=
x
3
+3
x
2

3
x
.
a) At any point (
x
0
,y
0
) on the graph, what is the slope of the tangent line to the graph?
b) The graph of
f
(
x
) has two tangent lines parallel to the line
y
=6
x
+ 100. Find the equations of
these two lines.
2.
a) Find all points on the graph of
y
=
x
2
whose tangent lines pass through the point (5
,
0).
b) Show that no line tangent to the graph of
f
(
x
)=
x
+
1
x
passes through the origin.
3. In worksheet 5, you used the Squeeze Theorem to show that
lim
θ
→
0
sin
θ
θ
=1
.
(1)
Recall that in the process of proving that (sin
θ
)
0
=cos
θ
we used limit (1) and
lim
θ
→
0
1

cos
θ
θ
=0
.
(2)
The goal of this problem is to prove limit (2). The diagram below will be helpful.
1
OA
Θ
B
P
a) We will only consider approaching 0 through positive angles, that is we are computing a righthand
limit. Prove that this is suﬃcient by showing that the lefthand limit must be the same.
(Hint: is cosine an even or odd function?)
b) Triangle
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 Spring '08
 Grether
 Equations, Slope

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