Md. Farooq Hasan
Lecturer (BUBT)
Assignment on Differential Calculus
1. Find the domain and range of the following functions:
i.
𝑓(𝑥) = 𝑥 2 + 2𝑥 − 1.
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ii.
𝑔(𝑥) =
iii.
ℎ(𝑥) = ln(𝑥 − 2)
iv.
𝑙(𝑥) = 𝑥 2 −3𝑥+2
√𝑥−6
𝑥2
2. Find the domain and range of the function f ( x) 
one-one and find f
1
( x) .
3. Find the domain and range of the function f ( x) 
one-one and find f
1
x5
. Also show that f (x) is
2x  3
5x  7
. Also show that f (x) is
7x  2
( x) .
𝑥2 ;
𝑥<1
4. Investigate the continuity of the function, 𝑓(𝑥) = { 2.5 ;
𝑥 = 1 , at 𝑥 = 1 .
2
𝑥 +5; 𝑥 >1
𝑥2 + 1 ;
𝑥<0
𝑥
;
0
≤
𝑥 < 1 ,at 𝑥 = 0
5. Investigate the continuity of the function, 𝑓(𝑥) = {
1
;
𝑥≥1
𝑥
and 𝑥 = 1 .
3
3 + 2𝑥; − 2 ≤ 𝑥 < 0
6. Investigate the continuity of the function, 𝑔(𝑥) =
3 − 2𝑥;
{−3 − 2𝑥;
and 𝑥 = 3/2 .
3
0 ≤ 𝑥 < 2 , at 𝑥 = 0
3
𝑥≥2
7. Find the first derivative of the followings with respect to 𝑥:
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i.
1 x 5
ln 

1 x 
ii.
iii.
cos ec 2 ( x 2  3 ) .
iv.
8. Find
i.
dy
of the followings:
dx
y  xx
ii.
𝑦 = (𝑠𝑖𝑛𝑥)𝑐𝑜𝑠𝑥
iii.
𝑦 = tan[𝑙𝑛𝑠𝑖𝑛(𝑒 𝑥 )]
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( x 2  1) sin 1 x  e
1−𝑥
𝑡𝑎𝑛−1 √1+𝑥
1 2 x
Md. Farooq Hasan
Lecturer (BUBT)
2
2
iv.
v.
𝑦 = 𝑒 𝑐𝑜𝑠𝑒𝑐 √𝑥 +3
x 2  2 xy  y 2  5
vi.
vii.
viii.
𝑦 = ln(𝑠𝑖𝑛2 𝑥 2 )
𝑥 2 + 𝑦 2 = 2𝑎𝑥𝑦
𝑦 = 𝑒 𝑡 𝑠𝑖𝑛𝑡 ,
𝑥 = 𝑒 𝑡 𝑐𝑜𝑠𝑡
ix.
𝑡𝑎𝑛𝑦 = 1−𝑡 2 ,
x.
𝑥 = 𝑎(𝜃 + 𝑠𝑖𝑛𝜃) ,
2𝑡
2𝑡
𝑥 = 1+𝑡 2
𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝜃)
9. Find the maximum and minimum values of the function f ( x)  x 3  6 x 2  9 x  5 .
10. Find the maximum and minimum values of the function f ( x)  2 x 3  6 x 2  18x  7 .
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11. Show that the maximum value of the function 𝑓(𝑥) = 𝑥 + 𝑥 is less than its minimum
value.
12. If 𝑦 = 𝑠𝑖𝑛−1 𝑥 , then by using the Leibnitz’s theorem show that
(1  x 2 ) y n2  (2n  1) xy n1  n 2 y n  0.
13. If y  cos 1 x , then by using the Leibnitz’s theorem show that
(1  x 2 ) y n2  (2n  1) xy n1  n 2 y n  0.
14. The volume of a spherical balloon is increasing at the rate of 12 cm3/sec. Find the
rate of change of its surface at the instant when its radius is 6 cm.
15. The volume of a spherical balloon is decreasing at the rate of 1000 cm3/sec. Find the
rate of change of its surface at the instant when its radius is 10 cm.
16. A garden is to be laid out in a rectangular area and protected by a chicken wire fence.
What is the largest possible area of the garden if only 100 running feet of chicken wire
is available for the fence?
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