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.