Differential and Integral Calculus

Daffodil International University Department of Electrical and Electronic Engineering Faculty of Engineering Midterm Examination, Fall-2026 Course Code: 0541-111 Course Title: Differential and Integral Calculus Section: A, B, C, D, R Level-Term: L1-T1 Teacher's Initial: SDN Full Marks: 25 Exam Date: Oct 31, 2026 Time: 1.5 Hours [Answer all the following questions sequentially] [The second right-most column indicates the COs with the respective Bloom's level] [The right-most column indicates the assigned marks] Q1. (a) Describe the Mean Value Theorem with its physical meaning. | 3 (b) A function is defined as follows f(x) = { x^2 + 1, when x < 0 x, when 0 ≤ x ≤ 1 1/x, when x > 1 (i) Discuss the continuity and differentiability at x = 0. | CO-1 C(2) | 5 (ii) Describe your findings graphically. (c) Identify the horizontal asymptotes of the Sigmoid function σ(x) = 1 / (1 + e^-x). | 2 Q2. (a) Determine y_n for the function y = ln(ax + b) | 2 (b) An equation used in thermodynamics is the Benedict-Webb-Rubine equation of state for the expansion of a gas. The equation is: p = RT/V + (B_0 RT - A_0 - C_0/T^2)/V^2 + (bRT - a)/V^3 + Aα/V^6 + (C(1 + γ/V^2)/T^2)(1/V^3)e^(-γ/V^2) show that ∂^2p/∂T^2 = 6/(V^2 T^4) {(1 + γ/V^2)e^(-γ/V^2) - c_0}. | CO-2 C(3) | 3 (c) Determine the n-th derivative of the function y = (1 + x^2)e^(2x) using Leibniz theorem. | 2 Q3. (a) A capacitor in an RC circuit discharges according to v_C(t) = V_0 e^(-t / RC) where V_0 = 12 V, R = 1 kohm, C = 1 muF. (i) Construct a fourth-order Maclaurin polynomial. (ii) Calculate the percentage error of v_C(t) at = 0.5 ms. | CO-3 C(3) | 5 (iii) Calculate the fourth-order Lagrange's form of the remainder. (b) The output voltage of a temperature sensor is given by V(T) = 0.001T^2 + 0.5T + 1. Calculate the temperature T at which the average rate of change of the sensor voltage occurs over the interval [20, 80]°C. | 3