Formula sheet for Final Exam in Math 106 on Monday 14, 2007
Derviative formulas
(
x
p
)
±
=
px
p

1
(
e
x
)
±
=
e
x
(ln
x
)
±
=
1
x
General properties of derivatives
(
cf
)
±
=
cf
±
(
f
+
g
)
±
=
f
±
+
g
±
(
fg
)
±
=
f
±
g
+
±
(1
/g
)
±
=

g
±
g
2
(
f/g
)
±
=
f
±
g

±
g
2
(
f
(
g
))
±
=
f
±
(
g
)
g
±
Equation to the tangent line to the graph at
x
0
:
y
=
f
(
x
0
)+
f
±
(
x
0
)(
x

x
0
)
Exponentials.
lim
n
→∞
(
1+
1
n
)
n
=
e
=2
.
71828. Properties:
x
p
x
q
=
x
p
+
q
e
x
e
y
=
e
x
+
y
1
/x
p
=
x

p
1
/e
x
=
e

x
(
x
p
)
q
=
x
pq
(
e
x
)
y
=
e
xy
h
(
t
)=
Ae
rt
satisfes
h
±
(
t
rh
(
t
),
h
(0) =
A
.I
F
r>
0,
t
= (ln(2))
/r
is the
doubling time
.
IF
r<
0,
t
= (ln(1
/
2))
/r
is the
halflife
.
Natural Logarithm.
By defnition
e
ln
x
=
x
. This implies ln(
e
x
x
. Properties:
ln(
uv
) = ln(
u
)+ln(
v
)
ln(1
/u

ln(
u
)
ln(
u/v
)=ln(
u
)

ln(
v
)
ln(
u
p
p
ln(
u
)
Maxima and minima, increasing, decreasing, convex, concave
f
is increasing where
f
±
(
x
)
>
0, decreasing where
f
±
(
x
)
<
0.
IF
f
±
(
c
) = 0 then
c
is a critical point. (This diﬀers From the book, but is the commonly
accepted defnition). This is a necessary condition For the point
c
to be a local maximum
[minimum], i.e., there are
a<c<b
so that
f
(
c
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 Spring '07
 DURRETT
 Calculus, Derivative, Formulas, dx

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