INDEPENDENT UNIVERSITY, BANGLADESH (IUB)
MAT 301 Midterm
Total marks: 30 Time: 60 minutes
Answer any five (5)
1. Lets, 𝑓(𝑥) = 𝑥 4 − 5𝑥 3 + 9𝑥 2
a) Find all critical points and infection points, if any.
b) The intervals on which 𝑓 is increasing and those on which 𝑓 is decreasing.
c) The intervals on which 𝑓 is concave up and those on which 𝑓 is concave down.
𝑥 2 −7𝑥+6
2. Lets, 𝑓(𝑥) = 𝑥−10
a) Find all critical points, if any.
b) The classification of each critical point, if any, as a minimum or maximum, local or global, or
not an extremum.
c) Also, find the local and absolute extrema.
𝑑𝑥
3. (a) Evaluate ∫ 1+𝑠𝑖𝑛 𝑥.
1
(b) Evaluate ∫0
𝑑𝑥
.
𝑥 2 −6𝑥+10
4. (a) Solve the initial value problems,
1
𝑑𝑦
𝑑𝑡
𝜋
1
= 𝑠𝑖𝑛 𝑡 + 1 ; 𝑦 ( 3 ) = 2.
(b) Evaluate ∫0 √1 − 𝑥 2 𝑑𝑥.
5. Let, 𝑓(𝑥, 𝑦) = √5𝑥 5 𝑦 − 𝑥 2 𝑦 3
(a) Find the slope of the surface 𝑧 = 𝑓(𝑥, 𝑦) in the 𝑥 −direction at the point (1, 1).
(b) Find the slope of the surface 𝑧 = 𝑓(𝑥, 𝑦) in the 𝑦 −direction at the point (1, 1).
𝑑𝑧
6. (a) Find 𝑑𝑡 , where 𝑧 = ln(2𝑥 2 + 𝑦) , 𝑥 = √𝑡, 𝑦 = 𝑡 2/3 .
𝜕𝑧
𝜕𝑧
(b) Find 𝜕𝑢 and 𝜕𝑣, where 𝑧 = 8𝑥 2 + 2𝑥 − 3𝑦, 𝑥 = 𝑢𝑣, 𝑦 = 𝑢/𝑣.