Calculus and Ordinary Differential Equation

Daffodil International University B. Sc. in Information and Communication Engineering Mid Term Examination, Fall 2026 Course Code: MAT 1031 Course Title: Calculus and Ordinary Differential Equation Section: A Teacher's Initial: MN Full Marks: 25 Date: 31 Oct 2026 Time: 1.5 Hours Note: Answer all of the following questions. Right hand margin indicates full marks. 1. Define odd and even function. Identify whether the following functions are odd or even: (CLO1, C3) [1+1.5+2.5] (i) sin^3(x) · cos^3(x) (ii) ln (x + sqrt(x^2 + 1)) 2. Sketch the graph of the following functions: (CLO1, C3) [02+02+02] (i) y = |x - 3| + 2 (ii) y = sqrt(x + 3) (iii) y = (x + 2)^2 3. Check the continuity of the function f(x) at x = 0 and x = pi/2: (CLO1, C3) [04] f(x) = { 1, -inf < x < 0 1 + sin x, 0 <= x <= pi/2 2 + (x - pi/2)^2, pi/2 < x < inf 4. Demonstrate the following problems (any two): (CLO2, C2) [02+02] (i) If y = ln (x + sqrt(x^2 + a^2)), then show that y' = 1/sqrt(x^2 + a^2) (ii) If y = ln(sec x + tan x), then show that y' = sec x (iii) If y = x^n, then find y_n (n-th derivative of y) 5. Demonstrate the following problems: (CLO2, C2) [03+03] (i) If f(x) = x^3/3 + ax^2 - 3a^2x, then find the maximum and minimum values of f(x). (ii) If u = sin^-1((x + y)/(sqrt(x) + sqrt(y))), then show that x(du/dx) + y(du/dy) = 1/2 tan u by applying Euler's Theorem on Homogeneous Function.