PROJECT WORTH 50 POINTS –
1) NO LATE SUBMISSIONS WILL BE ACCEPTED
2) COMPLETED PROJECTS NEED TO BE LEGIBLE
I. Computing Derivatives (slope of curve at a point) of polynomial functions.
For each of the following functions in a.-e. below perform the following three steps:
1. compute the
difference quotient
2. simplify expression from part 1. such that h has been canceled from the denominator
3. substitute and simplify
a.
b.
c.
d.
e. consider , using the results from parts a. through d.,
f. find a
general formula for (steps 1 through 3 performed).
II. Show that
Consider the
unit circle with in
standard position in
QI.
a. show that the
area of the right triangle (see diagram) is
(1,tanϴ)
ϴ
1
b. show that the
area of the sector (see diagram) is
ϴ
1
c. show that the
area of the acute triangle (see diagram)
(cos ϴ,sinϴ)
ϴ
1
d. set up the
inequality
e. multiply the
inequality in part d. by . (direction of inequalities is unchanged)
f. take the reciprocal of each term from part e. The direction of the inequality must be reversed because .
g. plug in 0 for for only. The result should be
III. Show that
a. multiply by
b. use trigonometric identity to
rewrite the numerator of the expression in part a. in terms of
c.
factor the expression in part b. with
one factor equal to . (find remaining factor).
d. use the fact that and substitute in the second factor (result is 0)
IV. Show that derivative of
a. find the
difference quotient for
(use sum angle formula )
b.
factor out of the
two terms in the
numerator with in part a
c. split up the expression in part b with each term over the denominator h
d. use identities to simplify part c. to
Thus you have shown that if .
0
=
h
c
x
f
=
)
(
b
ax
x
f
+
=
)
(
c
bx
ax
x
f
+
+
=
2
)
(
d
cx
bx
ax
x
f
+
+
+
=
2
3
)
(
(
)
3
2
2
3
3
3
3
:
int
h
xh
h
x
x
h
x
H
+
+
+
=
+
0
1
1
1
.....
)
(
a
x
a
x
a
x
a
x
f
n
n
n
n
+
+
+
+
=
-
-
0
)
(
)
(
)
(
®
-
+
=
h
as
h
x
f
h
x
f
x
f
0
1
sin
®
=
q
q
q
as
q
2
tan
q
=
Triangle
Right
Area
2
q
=
Sector
Area
2
sin
q
=
Triangle
Acute
Area
2
sin
2
2
tan
q
q
q
³
³
q
sin
2
b
a
b
a
b
a
if
1
1
0
,
<
®
<
®
>
q
q
cos
0
1
sin
1
®
£
£
q
q
q
as
0
0
cos
1
®
=
-
q
q
q
as
q
q
cos
1
-
q
q
cos
1
cos
1
+
+
1
sin
cos
2
2
=
+
q
q
q
2
sin
q
q
sin
0
1
sin
®
=
q
q
q
as
0
=
q
q
q
cos
sin
=
h
f
h
f
f
)
(
)
(
)
(
q
q
q
-
+
=
(
)
q
q
sin
=
f
(
)
h
h
h
sin
cos
cos
sin
sin
q
q
q
+
=
+
q
sin
q
sin
0
0
cos
1
1
sin
®
=
-
=
h
as
h
h
and
h
h
q
cos
h
x
f
h
x
f
x
f
)
(
)
(
)
(
-
+
=
q
q
q
q
q
q
cos
0
)
(
)
(
)
(
,
sin
)
(
=
®
-
+
=
=
h
as
h
f
h
f
f
then
f
Taha Gursoy
m166calculusprojectfall20151.docx
0
