LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
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1
LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
CH-5
(CONTINUITY AND DIFFERENTIABILITY)
Formula used while calculating limits:(1) If lim
(2) lim
→
(3) lim
→
(4) lim
→
(5) lim
→
= 𝑛𝑎
→
(6) lim
→
(7) lim
→
=1
(8) lim
→
= log 𝑎
(9) lim
→
=1
cos 𝑥 = 1
=1
cos(𝑥 − 𝑎) = 1
(
(
(10) lim →
=1
)
)
=1
=1
Some trigonometric functions
(i)
1 + cos 2𝑥 = 2 𝑐𝑜𝑠 𝑥
(iv) 1 − cos 𝑥 = 2 𝑠𝑖𝑛
(ii)
1 + cos 2𝑥 = 2 𝑐𝑜𝑠
(v) sin 2𝑥 = 2 sin 𝑥 cos 𝑥
(iii)
1 − cos 2𝑥 = 2 𝑠𝑖𝑛 𝑥
(vi) sin 𝑥 = 2 sin cos
Continuous Functions:
1) Y = 3𝑥 + 1
4) F(x) =
6) 𝑓 (𝑥) =
2) Y = 𝑥 + 2
𝑥 sin
𝑥≠0
0 𝑥=0
𝑥≠
1 𝑥=
⎧ − 𝑥 , 0 ≤ 𝑥 < 1/2
⎪
1, 𝑥 =
8)∶ 𝑓(𝑥) =
⎨
⎪ −𝑥 , <𝑥 ≤0
⎩
3) Y = sin 𝑥
+ cos 𝑥 𝑥 ≠ 0
5) 𝑓 (𝑥) =
7) 𝑓 (𝑥) =
2 𝑥=0
5𝑥 − 4 0 < 𝑥 ≤ 1
4𝑥 − 3𝑥 1 < 𝑥 < 2
9) : 𝑓 (𝑥) =
2
2𝑥 − 1 𝑥 < 0
2𝑥 + 1 𝑥 ≥ 0
LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
10)∶ 𝑓(𝑥) =
𝑥+2 𝑥 ≤1
𝑥−1 𝑥 >1
11) 𝑓 (𝑥) =
|𝑥| + 3 𝑥 ≤ −3
(
)
12) : 𝑓 𝑥 = −2𝑥 − 3 < 𝑥 > 1
6𝑥 + 3 𝑥 > 3
| |
𝑥≠0
1 𝑥=0
13) 𝑓 (𝑥) =
2𝑥 + 3 𝑥 < 0
0 𝑥=0
𝑥 +3 𝑥 >0
𝑥−1 𝑥 <0
14) 𝑓 (𝑥) = 1/4 𝑥 = 0 can be made continuous by f(0)= −1.
𝑥 −1 𝑥 > 0
Answers : Questions 1,2,3,4,5,6,7 are continuous.
Q : 8, 9, 10, 11 and 13 are not continuous, Q 12 : Continuous at 𝑥 = −3 but not
continuous at 𝑥 = 3.
Topic : To find the value of some constant when the given f(x) is continuous.
15) 𝑓(𝑥) =
𝑥 ≠ −1
(𝐴𝑛𝑠: 𝑘 = 6)
𝑘 𝑥=1
16) 𝑓 (𝑥) =
𝑥≠0
𝑘 𝑥=0
(𝐴𝑛𝑠: 𝑘 = 1)
3ax + b 𝑥 > 1
17) 𝑓(𝑥) =
11 𝑥 = 1
5𝑎𝑥 − 2𝑏 𝑥 < 1
(𝐴𝑛𝑠: 𝑎 = 3, 𝑏 = 2)
1 𝑥≤3
18) 𝑓(𝑥) = 𝑎𝑥 + 𝑏 3 < 𝑥 ≤ 5
7
5≤𝑥
(𝐴𝑛𝑠: 𝑎 = 3, 𝑏 = −8)
𝑥 + a√2 sin 𝑥
0 ≤ 𝑥 < 𝜋 ⁄4
19) 𝑓(𝑥) = 2𝑥 × 𝑐𝑜𝑡𝑥 + 𝑏 𝜋⁄4 ≤ 𝑥 < 𝜋⁄2 (𝐴𝑛𝑠: 𝑎 = 𝜋⁄6 , 𝑏 = − 𝜋⁄12)
acos 2𝑥 − b sin 𝑥 𝜋⁄2 ≤ 𝑥 < 𝜋
20) 𝑓(𝑥) =
𝑥² + a𝑥 + b
0≤𝑥<2
3𝑥 + 2 2 ≤ 𝑥 ≤ 4
2a𝑥 + 5b 4 < 𝑥 ≤ 8
(𝐴𝑛𝑠: 𝑎 = 3, 𝑏 = −2)
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LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
21) 𝑓 (𝑡 ) =
− 𝑥 0 ≤ 𝑥 ≤ 1 ⁄2
𝑡 ≠ 𝜋 ⁄2
⁄ –
22) 𝑓 (𝑥) =
1 𝑡 = 𝜋 ⁄2
1 𝑥 = 1 ⁄2
− 𝑥 1/2 < 𝑥 ≤ 1
21) 𝐴𝑛𝑠; LHL = RHL = 1
22) 𝐴𝑛𝑠; LHL = 0 RHL = 1
3𝑎𝑥 + 𝑏 𝑥 > 1
23) 𝐼𝑓 𝑡ℎ𝑒 𝑓 (𝑥) =
11 𝑥 = 1
5𝑎𝑥 − 2𝑏 𝑥 < 1
𝐴𝑛𝑠; 𝑎 = 3, 𝑏 = 2
is continuous at 𝑥 = 1. ind a and b?
24) Discuss the continuity of 𝑓 (𝑥)𝑎𝑡 𝑥 = 0, 𝑓(𝑥) =
2𝑥 − 1 𝑥 < 0
2𝑥 + 1 𝑥 ≥ 0
{𝐴𝑛𝑠: 𝐿𝐻𝐿 = −1 𝑅𝐻𝐿 = 1}
25) Show that 𝑓(𝑥) = 2𝑥 − |𝑥 | is continuous at 𝑥 = 0. {𝐿𝐻𝐿 = 𝑅𝐻𝐿 = 0}
𝑥≠0
26) For what value of a, 𝑓(𝑥) =
𝑘
𝑥=0
⎧
27) For what value of a, 𝑓(𝑥) =
𝑥≠2
28) If 𝑓(𝑥) =
𝑎
⎨
⎩
√
√
is continuous. (𝐴𝑛𝑠: 𝑘 = 1)
𝑥<0 ⎫
𝑥=0
is continuous. (𝐴𝑛𝑠: 𝑎 = 8)
⎬
𝑥 > 0⎭
is continuous at 𝑥 = 2. Find K. 𝐴𝑛𝑠 𝑘 =
𝑘 𝑥=2
𝐹𝑖𝑛𝑑 𝑎 𝑎𝑛𝑑 𝑏 𝑓𝑜𝑟 𝑓(𝑥) 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑜𝑛 [0, 𝜋].
29) 𝑓 (𝑥) =
𝑥 + 𝑎√2 𝑠𝑖𝑛𝑥 0 ≤ 𝑥 ≤ 𝜋⁄4
2𝑥 𝑐𝑜𝑡𝑥 + 𝑏
≤ 𝑥 ≤ 𝜋 ⁄2
𝑎 cos 2𝑥 − 𝑏 𝑠𝑖𝑛𝑥
𝐴𝑛𝑠; 𝑎 =
,𝑏 = −
≤𝑥≤𝜋
DIFFERENTIABILITY
1)
2)
3)
4)
Show that 𝑓(𝑥) = 𝑥 is differentiable at 𝑥 = 1.
Show that 𝑓(𝑥) = |𝑥 | is not differentiable at 𝑥 = 0.
Show that 𝑓(𝑥) = |𝑥 − 3| is not differentiable at 𝑥 = 3.
Show that 𝑓(𝑥) = | 𝑥 + 1| + |𝑥 − 1| is not differentiable at 𝑥 = −1.
𝑥 − 1, 𝑥 < 2
5) Show that 𝑓(𝑥) =
is differentiable at 𝑥 = 2.
2𝑥 − 3, 𝑥 ≥ 2
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LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
3𝑥 − 2 0 < 𝑥 ≤ 1
6) Check continuity and differentiability of 𝑓(𝑥) = 2𝑥 − 𝑥 1 < 𝑥 ≤ 2 at
5𝑥 − 4 𝑥 > 2
𝑥 = 1, 𝑥 = 2.
7) Find the values of a and b if 𝑓(𝑥) = 𝑥 + 3𝑥 + 𝑎 𝑥 ≤ 1 is differentiable at
𝑏𝑥 + 2 𝑥 > 1
𝑥 = 1.
8) Discuss the continuity and differentiability of
1−𝑥 𝑥 <1
𝑓(𝑥) = (1 − 𝑥)(2 − 𝑥) 1 ≤ 𝑥 ≤ 2
𝑎𝑡 𝑥 = 1, 𝑥 = 2.
3−𝑥 𝑥 >2
Answers : 7) 𝑎 = 3, 𝑏 = 5
8) Continuous and differentiable at 𝑥 = 1 but not continuous and differentiable at
𝑥 = 2.
Differentiation:
Formula used
(1)
𝑥 =𝑛𝑥
(2)
𝑎 = 𝑎 𝑙𝑜𝑔 𝑎
(4)
tan 𝑥 = 𝑠𝑒𝑐 𝑥
(5)
sec 𝑥 = sec 𝑥 tan 𝑥 (6)
(7)
log 𝑥 =
(8)
cos 𝑥 = − sin 𝑥
(10)
(3)
(9)
sin 𝑥 = cos 𝑥
𝑒 = 𝑒
cot 𝑥 = − cosec 𝑥
cosec 𝑥 = − cosec 𝑥 cot 𝑥
Notes
(i) Differentiation of a constant function, like
(ii)
𝑐𝑓 (𝑥) = 𝑐
(𝑘) = 0
𝑓 (𝑥), 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
(iii) 𝑷𝒓𝒐𝒅𝒖𝒄𝒕 𝑹𝒖𝒍𝒆: 𝒍𝒆𝒕 𝒈(𝒙) 𝒂𝒏𝒅 𝒇(𝒙) 𝒂𝒓𝒆 𝒕𝒘𝒐 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔, 𝒕𝒉𝒆𝒏
𝑔 (𝑥 ) 𝑓 (𝑥 ) = 𝑔 (𝑥 ).
(iv) Generalization of
v) Quotient Rule:
𝑓 (𝑥 ) + 𝑓 (𝑥 ) .
𝑔 (𝑥 )
𝑔(𝑥). 𝑓(𝑥). ℎ(𝑥) = 𝑓(𝑥). 𝑔(𝑥).
( )
( )
=
( ).
( )
( ).
ℎ(𝑥) + ℎ(𝑥). 𝑓(𝑥).
( )
[ ( )]
5
𝑔(𝑥) + 𝑔(𝑥). ℎ(𝑥)
𝑑
𝑓(𝑥)
𝑑𝑥
LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
some formulae used in 1st principal method of differentiation:
(1) If lim
(2) lim
→
(3) lim
→
(4) lim
→
(5) lim
→
= 𝑛𝑎
→
(6) lim
= 𝑎,
=1
= 𝑙𝑜𝑔 𝑎
(
)
(
(7) lim
→
(8) lim
→
(9) lim
→
)
=1
=1
(
(10) lim →
= 𝑙𝑜𝑔 𝑎
= 𝑎, lim
→
)
=1
=1
Trigonometric formulae used in differentiation
(i) cos 2𝜃 = cos 𝜃 − sin 𝜃 = 1 − 2 sin 𝜃 = 2 cos 𝜃 − 1 =
(ii) sin 𝑥 = 2 sin cos =
, sin 2𝑥 = 2 sin 𝑥 cos 𝑥 =
(iii) 1 + cos 2𝑥 = 2 cos 𝑥 , 1 + cos 𝑥 = 2 cos
(iv)
1 − cos 2𝑥 = 2 sin 𝑥 , 1 − cos 𝑥 = 2 sin
(v)
tan 2𝑥 =
(vi)
(vii)
cos 3𝑥 = 4 cos³ 𝑥 − 3 cos 𝑥 , sin 3𝑥 = 3 sin 𝑥 − 4 sin 𝑥
1 ± sin 𝑥 = (sin ± cos )
(viii)
1 ± sin 2𝑥 = (sin 𝑥 ± cos 𝑥)
(ix)
tan(x ∓ y) =
(x)
𝒄𝒐𝒔𝜽 𝒔𝒊𝒏𝜽
𝒄𝒐𝒔𝜽 𝒔𝒊𝒏𝜽
³
, tan 3𝑥 =
=
∓
±
𝟏 𝒕𝒂𝒏𝜽
𝟏 𝒕𝒂𝒏𝜽
= 𝒕𝒂𝒏(𝝅𝟒 𝜽)
Some important substitutions:
(i) 𝑎 + 𝑥
𝑥 = 𝑎 𝑡𝑎𝑛𝜃, 𝑎 𝑐𝑜𝑡𝜃 , 𝑎 − 𝑥 , 𝑥 = 𝑎 𝑠𝑖𝑛𝜃, 𝑎 𝑐𝑜𝑠𝜃
(ii) 𝑎 − 𝑥 , 𝑥 = 𝑎 𝑠𝑒𝑐𝜃, 𝑎 𝑐𝑜𝑠𝑒𝑐𝜃
(iii)
𝑜𝑟
, 𝑥 = 𝑎𝑐𝑜𝑠2𝜃
𝑜𝑟
6
, 𝑥 = 𝑎 𝑐𝑜𝑠2𝜃
(
→
)
=1
LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
(iv)1 + 𝑥 => 𝑥 = 𝑡𝑎𝑛𝜃, 𝑐𝑜𝑡𝜃 , 1 + tan 𝜃 = sec 𝜃 , 1 + cot 𝜃 = 𝑐𝑜𝑠𝑒𝑐 𝜃
(v) 1 − 𝑥 => 𝑥 = 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃 , 1 − sin 𝜃 = cos 𝜃 , 1 + cos 𝜃 = sin 𝜃
(vi) 𝑥 − 1 => 𝑥 = 𝑠𝑒𝑐𝜃, 𝑐𝑜𝑠𝑒𝑐𝜃 , sec 𝜃 − 1 = 𝑡𝑎𝑛 𝜃
(vii)
=> 𝑥 = 𝑐𝑜𝑠2𝜃
Topic 1: Basic Differentiation
1. (i)
(ii) 𝑥
(iii)
(iv) √𝑥
2. (i) 𝑒
(ii) 𝑥. 𝑒
(iii) 𝑥. 𝑐𝑜𝑠𝑥
3. (i)
(ii)
√
(iv) 𝑥. log 𝑥
(iii) 𝑒 (𝑥 + 𝑙𝑜𝑔𝑥)
4. (i) (𝑥 + cos 𝑥 )(𝑥 − tan 𝑥)
(ii) 𝑥 sin 𝑥 log 𝑥
5. (i) 𝑥 𝑒 cos 𝑥
(𝑠 + 3𝑥)
6. (i)
(v)
(ii) 𝑥
(ii)
7. (i)
(ii)
(iii)
8. (i)
(ii)
(iii)
Topic 2: Derivative of a composite functions: {also known as a chain rule}
If 𝑧 = 𝑓(𝑦) 𝑎𝑛𝑑 𝑦 = 𝑔(𝑥) then
1) sin 2𝑥
6) sec 𝑥
.
2) cos 3𝑥
3) tan 4𝑥
4) sin(3𝑥 + 5) 5) sin(𝑥 + 3)
7) sin 𝑥
8) cos 𝑥
9) tan 𝑥
12) √cot 𝑥
13) (3𝑥 + 5)
16) cos √𝑥
17) 𝑒
22) 𝑒
=
11)√𝑐𝑜𝑠2𝑥
14) (3𝑥 + 4𝑥 + 5) 15) sec 𝑥
18) 𝑒
23) sin(𝑒 )
10) √sin 𝑥
19) 𝑒
24) √𝑥 + 𝑥 + 1
7
20) 𝑒
21) 𝑒 √
25) log sin 𝑥
LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
26) 𝑦 = log sin 𝑥
27) 𝑦 = 𝑠𝑖𝑛 𝑥 28) 𝑦 = sec tan √𝑥
29) 𝑦 = cos 𝑥 . sin (𝑥 ) 30) 𝑦 = sin(cos(𝑥 )) 31) log tan
32) log sin
33) 𝑒
−1
34) 𝑠𝑖𝑛 𝑥
+
35) 𝑒
36) sec (log 𝑥) 37) log(𝑥 + √𝑎 + 𝑥 ) 38) 𝑙𝑜𝑔 𝑙𝑜𝑔 𝑥 39) 𝑙𝑜𝑔 2
40) tan(𝑒
) 41) 𝑙𝑜𝑔
44)
45) 3
√
42) 𝑒
46) 3
47) 2
50) √sin 𝑥 51) √tan 𝑥
52) 𝑒
53) 𝑦 = 𝑥 + √𝑥 + 𝑎
, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
54) 𝑦 =
, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 (1 − 𝑥 )
55) 𝑦 = √𝑎 − 𝑥 −
𝑠𝑖𝑛
log 𝑠𝑖𝑛2𝑥 43) (sin
𝑥 )
48) sin 𝑥 49) cos 𝑥
=√
+𝑦 =0
, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
= √𝑎 − 𝑥
Topic 3: Derivative of implicit Functions
1) 𝑦 + sin 𝑦 = cos 𝑥
2) 𝑥 + 2𝑥𝑦 + 𝑦 = 42 3) 𝑥 + 𝑥 𝑦 + 𝑦 𝑥 + 𝑦 = 81
4) 𝑥 1 + 𝑦 + 𝑦√1 + 𝑥 = 0 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
6) sin 𝑦 + cos 𝑥𝑦 = 𝜋 7) 𝑥𝑦 = 𝑐
9)
+
=
(
)
5) sin 𝑥 + cos 𝑦 = 1
8) 𝑦 − 3𝑥𝑦 = 𝑥 + 3𝑥 𝑦
= 1 10) tan (𝑥 + 𝑦 ) = 𝑎 11) sin(𝑥𝑦) + cos (𝑥 + 𝑦) = 1
12) (𝑥 + 𝑦 ) = 𝑥𝑦 13) sin 𝑦 + cos 𝑥𝑦 = 𝑘 𝑎𝑡 𝑥 = 1, 𝑎𝑛𝑑 𝑦 =
14) √1 − 𝑥 + 1 − 𝑦 = 𝑎(𝑥 − 𝑦)𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
8
=
LIBERTY ACADEMY , C-13, DAKSHIN PURI , NEW DELHI 110062, MOBILE - 9716892533
15) 𝑥 1 + 𝑦 + 𝑦√1 + 𝑥 = 0 , 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 (1 + 𝑥 )
16)
9
+1=0