Differential Calculus
Topic Overview :
A .
limits and Functions
A. 1 . Theorems on limits
A. 2 . Direct substitution Method
A. 3 . Factorization Method
A. 4 .
Rationalization Method
A. 5 .
Infinity Method
A. 6 . L' Hospital's Rule
A.7 . Gille Sania 's Principle for limits
B. Asymptotes
B. l . Horizontal
Asymptotes
Vertical
B. 2 .
Asymptotes
B. 3 . Inclined Asymptotes
C .
Basic Differentiation
c. 1 . Derivatives of Functions at specific Values
c. 2 .
Implicit Differentiation
c. 3 .
Higher Derivatives
D . Basic Applications
D. I . Motion Problems
D. 2 .
Tangent & Normal lines
D. 3 .
Critical Points & points of Inflection
D. 4 .
Slope of a curve
E . Maxima and Minima Applications
F. Time Related Rates
G . Parametric Equation
H .
Curvature and Radius of curvature
I . Curvilinear Motion
J . Partial Differentiation and Applications
,A . LIMITS AND FUNCTIONS
A. l .
Theorems on limits
If f is
13 . a
polynomial :
"
Assume that lift tax ,
)
and ¥794 exist and that limtcx ) -
-
Ha)
any constant
c is .
Then ,
x →a
tim c =
c
1 ,
x→ a
lim x =a
The limit lim f- (x) -
-
M
2 .
x→ a X la
-
limlcfcxl) climflx) if and
only if the right hand limits and left hand
-
-
-
-
z .
x -
sa x→ a limits exist and are
equal to M :
limflx) limfcx) M
limlflxltglx))
=
limglx)
-
limfcx) I
-
¢
-
-
.
x → at
-
x→a xta x sa -
x sa-
s .
limltcx)g( ) ) x
= limfcx) .
limgcx) Suppose that f and
g are two functions such that
x la
Iim f- Cx) =L to and limglx )
-
x→ a x⇒a -
-
o
limflx) xta x→a
lim tht Ha
limglx) to then the limit f- Ix ) does not exist .
↳ = ,
lim
g Cx ) limg (x)
x 'a
g (x )
-
x sa
-
x sa-
x la
-
" "
7. limlflx) ) -
-
flimflx) ) ,
where n EN
x -
sa x→ a
"
8 .
lim x = an
x →a
limn f- Cx) limflx ) lim f- (x ) > o if n
q
= n
,
is
,
sa x→a even
x 'a -
x -
to . limllntlx)) -
-
lnflimflx ) ) ,
lim f- (x) > o
x →a x→ a x sa-
11 .
Squeeze Rule :
If ffx) Eg (x ) E h (x ) for all x in an
open
interval that contains a , except possibly at x=a
,
limtcx) =
limhlx) =L limgcx) =L
and , then
x -
sa x
-
ta x→a
12 . Composition Rule :
If f- (x) is continuous at x
-
- b and limglx) - b
x→a
then , limffglx) ) =
fllimgcx))
x sa-
x→a
, Sample Problems :
Example No 1 -
Example No 4 ,
Lim 721-52-1-6 him 3×3-2×+4
2- → I Ztl X -70 2 -
3×2-2×3
Solution : solution :
Lim z2t5zt6 lim (172+511)+6 him 3×3-2×+4
=
2- → I Zt I
2- → l Itt x -7N 2 -
3×2-2×3
=
6 ( answer) recall :
limflx) -
-
limfcx) =M
+
-
Example No 2 x→a x sa-
.
'
Lim x -
y
x -12 x' +2×-8 X =
999,999
say '
Solution : Lim 31999,999) -
21999,999)t4
2 3
Lim X
'
-
4 lim ( Xt 2) ( x -2) x -7999,999 2 -
31999,999) -
21999,999)
=
x' 1-2×-8 (Xt 4) ( X 2) 1.5
= -
x -12 X -72 -
lim xtz
=
X -72 Xt 4 X =
-999,999
say
lim 21-2 him 31-999,999/-21-999,999)t4
=
2
3
X -72 2+4 x -5999,999 2 -
31-999,999) -
21-999,999)
= -
1.5
=
f- } = ( answer )
: Lim 3×3-2×+4
( answer)
.
1.5
-
=
+ → a 2 3×2-2×3
Example No 3
-
,
Lim I
×
x→o
Solution :
recall :
Lim I )
x -70
×
limflx) -
-
limfcx) - M
t
-
x→a x sa-
say
X=
-
0.000001 ( left side of O )
^
Lim 1
X -7 0.000001 0.000001
- -
H it )
( ,
.
1,000,000
= -
-
V
X= 0.000001 ( right side of o )
say
I
Lim
X -70.000001 0.000001
= 1,000,000
#
since -1,000,000 ¥ 1,000,000 ,
- : LIMIT DOES NOT EXIST ( answer)
Topic Overview :
A .
limits and Functions
A. 1 . Theorems on limits
A. 2 . Direct substitution Method
A. 3 . Factorization Method
A. 4 .
Rationalization Method
A. 5 .
Infinity Method
A. 6 . L' Hospital's Rule
A.7 . Gille Sania 's Principle for limits
B. Asymptotes
B. l . Horizontal
Asymptotes
Vertical
B. 2 .
Asymptotes
B. 3 . Inclined Asymptotes
C .
Basic Differentiation
c. 1 . Derivatives of Functions at specific Values
c. 2 .
Implicit Differentiation
c. 3 .
Higher Derivatives
D . Basic Applications
D. I . Motion Problems
D. 2 .
Tangent & Normal lines
D. 3 .
Critical Points & points of Inflection
D. 4 .
Slope of a curve
E . Maxima and Minima Applications
F. Time Related Rates
G . Parametric Equation
H .
Curvature and Radius of curvature
I . Curvilinear Motion
J . Partial Differentiation and Applications
,A . LIMITS AND FUNCTIONS
A. l .
Theorems on limits
If f is
13 . a
polynomial :
"
Assume that lift tax ,
)
and ¥794 exist and that limtcx ) -
-
Ha)
any constant
c is .
Then ,
x →a
tim c =
c
1 ,
x→ a
lim x =a
The limit lim f- (x) -
-
M
2 .
x→ a X la
-
limlcfcxl) climflx) if and
only if the right hand limits and left hand
-
-
-
-
z .
x -
sa x→ a limits exist and are
equal to M :
limflx) limfcx) M
limlflxltglx))
=
limglx)
-
limfcx) I
-
¢
-
-
.
x → at
-
x→a xta x sa -
x sa-
s .
limltcx)g( ) ) x
= limfcx) .
limgcx) Suppose that f and
g are two functions such that
x la
Iim f- Cx) =L to and limglx )
-
x→ a x⇒a -
-
o
limflx) xta x→a
lim tht Ha
limglx) to then the limit f- Ix ) does not exist .
↳ = ,
lim
g Cx ) limg (x)
x 'a
g (x )
-
x sa
-
x sa-
x la
-
" "
7. limlflx) ) -
-
flimflx) ) ,
where n EN
x -
sa x→ a
"
8 .
lim x = an
x →a
limn f- Cx) limflx ) lim f- (x ) > o if n
q
= n
,
is
,
sa x→a even
x 'a -
x -
to . limllntlx)) -
-
lnflimflx ) ) ,
lim f- (x) > o
x →a x→ a x sa-
11 .
Squeeze Rule :
If ffx) Eg (x ) E h (x ) for all x in an
open
interval that contains a , except possibly at x=a
,
limtcx) =
limhlx) =L limgcx) =L
and , then
x -
sa x
-
ta x→a
12 . Composition Rule :
If f- (x) is continuous at x
-
- b and limglx) - b
x→a
then , limffglx) ) =
fllimgcx))
x sa-
x→a
, Sample Problems :
Example No 1 -
Example No 4 ,
Lim 721-52-1-6 him 3×3-2×+4
2- → I Ztl X -70 2 -
3×2-2×3
Solution : solution :
Lim z2t5zt6 lim (172+511)+6 him 3×3-2×+4
=
2- → I Zt I
2- → l Itt x -7N 2 -
3×2-2×3
=
6 ( answer) recall :
limflx) -
-
limfcx) =M
+
-
Example No 2 x→a x sa-
.
'
Lim x -
y
x -12 x' +2×-8 X =
999,999
say '
Solution : Lim 31999,999) -
21999,999)t4
2 3
Lim X
'
-
4 lim ( Xt 2) ( x -2) x -7999,999 2 -
31999,999) -
21999,999)
=
x' 1-2×-8 (Xt 4) ( X 2) 1.5
= -
x -12 X -72 -
lim xtz
=
X -72 Xt 4 X =
-999,999
say
lim 21-2 him 31-999,999/-21-999,999)t4
=
2
3
X -72 2+4 x -5999,999 2 -
31-999,999) -
21-999,999)
= -
1.5
=
f- } = ( answer )
: Lim 3×3-2×+4
( answer)
.
1.5
-
=
+ → a 2 3×2-2×3
Example No 3
-
,
Lim I
×
x→o
Solution :
recall :
Lim I )
x -70
×
limflx) -
-
limfcx) - M
t
-
x→a x sa-
say
X=
-
0.000001 ( left side of O )
^
Lim 1
X -7 0.000001 0.000001
- -
H it )
( ,
.
1,000,000
= -
-
V
X= 0.000001 ( right side of o )
say
I
Lim
X -70.000001 0.000001
= 1,000,000
#
since -1,000,000 ¥ 1,000,000 ,
- : LIMIT DOES NOT EXIST ( answer)
