Engineering Mathematics III Summative Assignment Question 1 a) Let 𝐶 be defined by, 𝒓(𝑡) = { (cos 𝑡 , sin 𝑡 , 0), 0 ≤ 𝑡 ≤ 1 3√3 (− 2 , 2 𝑡 (1 − 𝜋) , 0) , 2𝜋 3 2𝜋 3 ≤𝑡≤𝜋 Determine whether 𝒓(𝑡) is a smooth or discontinuous curve. [5] b) Find the arc-length of the curve defined by 𝑥 = 2𝑡, 𝑦 = 𝑡 2 and 𝑧 = ln 𝑡 , 𝑡 > 0, between the points (2,1,0) and (4,4, ln 2). [10] Question 2 a) Given 𝑤 = 𝑓(𝑥, 𝑦); 𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃; show that 𝜕𝑤 2 1 𝜕𝑤 2 𝜕𝑤 2 𝜕𝑤 2 ( 𝜕𝑟 ) + 𝑟 2 ( 𝜕𝜃 ) = ( 𝜕𝑥 ) + ( 𝜕𝑦 ) . [10] b) The voltage 𝑉 in a circuit that satisfies the law 𝑉 = 𝐼𝑅 is slowly dropping as the battery wears out. At the same time, the resistance 𝑅 is increasing as the resistor heats up. Use the equation 𝑑𝑉 𝜕𝑉 𝑑𝐼 𝜕𝑉 𝑑𝑅 = + 𝑑𝑡 𝜕𝐼 𝑑𝑡 𝜕𝑅 𝑑𝑡 to find how the current is changing at the instant when 𝑑𝑅 𝑅 = 600 𝑜ℎ𝑚𝑠, 𝐼 = 0.04𝑎𝑚𝑝, 𝑑𝑡 = 0.5 𝑜ℎ𝑚 𝑝𝑒𝑟 𝑠𝑒𝑐, and 𝑑𝑉 𝑑𝑡 = −0.01𝑣𝑜𝑙𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐. [5] Question 3 Consider the ordinary differential equation (ODE) 2𝑥𝑦 ′′ + (1 + 𝑥)𝑦 ′ + 3𝑦 = 0, in the neighbourhood of the origin. a) Show that 𝑥 = 0 is a regular singular point of the ODE. b) By seeking an appropriate solution to the ODE, show that [10] i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, 𝑐𝑛 , when one of the indicial equation roots is 1/2 is given by 𝑐1 = − (2𝑠+7)𝑐𝑠 7𝑐0 6 [10] 𝑐𝑠+1 = − (2𝑠+3)(𝑠+1) , 𝑠 = 1,2, … . . iii) For the indicial equation root 1/2, show that one of the solutions to the ODE is given by 𝑦 = 𝑐0 √𝑥 [1 − 7𝑥 6 + 21𝑥 2 40 − 77𝑥 3 560 +⋯] [5] Question 4 a) Using the Cauchy’s integral formula, evaluate 𝐼 = ∮𝐶 2𝑧 (1−3𝑧)(𝑧+2) 𝑑𝑧, where 𝐶 is a circle |𝑧| = 3. 10] 𝑧 2 +1 b) i) Find the residues of the function 𝑓(𝑧) = 𝑧(𝑧−6) at its poles in the finite complex plane. [5] ii)Using the residues in (4(b)(i), evaluate the integral ∮𝐶 𝑧 2 +1 𝑧(𝑧−6) 𝑑𝑧 where C is a contour that includes all the singularities of 𝑓(𝑧) and is in the positive sense. 2 [5]