Calculus I
Formula Sheet
Chapter 4
Section 4.1
1. Definition of the Extrema of a function:
Let
f
be defined on interval
I
:
•
( )
fc
is abs min when
( )
()fc fx≤
on
I
•
( )
fc
is abs max when
( )
()fc fx≥
on
I
2. Exterme Value Theorem:
If
f
is cts on
[,]ab
Then
f
has both max/min on
[,]ab
3. Definition of Relative Extrema:
• If
( )
fc
is max on
(,)ab
(open interval)
Then
( )
fc
is rel max
• If
( )
fc
is min on
(,)ab
(open interval)
Then
( )
fc
is rel min
4. Definition of a Critical Number:
Let
f
be defined at
c
Then
c
is a critical number if
o
()
0fc
′=
or
o
()
fc
′
DNE
5. Relative extrema occur only at c.n.
6. Find extrema on
[ , ]:ab
•
f
cts on
[,]ab
o Find c.n. on
(,)
ab
o Eval
f
at:
a
, all c.n.,
b
o Smallest = abs max
o Largest = abs min
Section 4.2
7. Rolle’s Theorem
•
f
cts on
[,]ab
•
f
diff on
(,)ab
•
( )
()f a fb=
⇒
there is at least one
c
in
(,)ab
such
that
( )
0fc
′=
8. Mean Value Theorem
•
f
cts on
[,]ab
•
f
diff on
(,)ab
⇒
there exists a
c
in
(,)ab
such that
( )
() ()fb fa
fc ba
−
′=−
Section 4.3
9. Definition of Increasing and Decreasing
• Increasing:
( ) ( )
12 1 2
x x fx fx<⇒ <
• Decreasing:
( ) ( )
12 1 2
x x fx fx<⇒ >
10. Test for Increasing and Decreasing
•
f
cts on
[,]ab
•
f
diff on
(,)ab
o
( )
0fx
′>
on
(,)ab
⇒
increasing
o
( )
0fx
′<
on
(,)ab
⇒
decreasing
o
( )
0fx
′=
on
(,)ab
⇒
constant
11. Find interval of increasing and decreasing
•
f
cts on
(,)ab
• Find c.n. on
(,)ab
• Create intervals
• Find the sign of
( )
fx
′
on each interval
o +
⇒
increasing
o
−⇒
decreasing
12. The First Derivative Test
•
c
is a c.n. in
(,)ab
•
f
cts on
(,)ab
•
f
diff on
(,)ab
except possibly at
c
o
( )
fc
′
change – to +
⇒
( )
fc
is rel min
o
( )
fc
′
change + to –
⇒
( )
fc
is rel max
o + to + or – to –
⇒
neither max nor min
