AP CALCULUS
SUMMER WORKSHOP III
Problems
:
1)
(
2004 Nova Scotia Math League
,
Game 2
:
Team Question
)
The parabola
y
x
=
2
meets the parabola
y
x
x
= −
+
−
2
6
4
at two points,
P
and
Q
.
Find the equation of the line passing through
P
and
Q
.
2)
(
2004 Nova Scotia Math League
,
Game 4
:
Team Question
) The integers
b
and
c
are chosen so that:
i)
The quadratic equation
5
0
2
x
bx
c
+
+
=
has two roots, one of which is 2.
ii)
The quadratic equation
5
0
2
x
cx
b
+
+
=
has two roots, one of which is 3.
Determine the values of
b
and
c
.
3)
(
2004 Nova Scotia Math League
,
Game 2
:
Team Question
)
In
Δ
ABC
, sides
AB
and
AC
are 10 units long.
Determine the maximum
possible area of
Δ
ABC
.
4)
Let
S
=
+
+
+
+
+
1
2
3
4
5
6
2009
2009
2009
2009
2009
2009
.
Determine the
last
digit of
S
.
Hint
:
Look for a
pattern
in each of the addends (i.e.
1
2
3
4
5
2009
2009
2009
2009
2009
,
,
,
,
and
6
2009
).
5)
Consider the function
(
)
f
x
x
x
=
−
2
2
2
.
a)
Find the domain of
(
)
f
x
.
b)
If
x
≥
1
, then find the equation of
the inverse function,
(
)
f
x
−
1
.
c)
Find the range of
(
)
f
x
.
6)
Find the domain of the following functions:
a)
(
)
f
x
x
x
=
+
−
4
b)
(
)
f
x
x
x
=
−
+
1
3
c)
(
)
(
)
f
x
x
=
−
ln 4
2
d)
(
)
f
x
x
x
x
=
−
+
+
2
7
10
1
7)
What is the range of
(
)
f
x
x
=
−
1
2
2
sin
?
Hint
:
Start with the inequality
− ≤
≤
1
2
1
sin
x
and perform some algebraic manipulations on it.
8)
Data was collected during an experiment in which an enzyme with its substrate was subjected to a medium with a
varying pH.
The result of the effects of pH on the action of the enzyme on its substrate is shown in the graph

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