questions 1,5= a= 12 b=1.2 c=0.1 questions 6,10 = g=0.6 h=400. k=-1.415

Differential Calculus In questions 1 to 5 you should use the values of a, b and c allocated to you in Table 1. 1) Differentiate a) y = 108'\" + 16x b)y = bt3 + 6sin(2t) 7 2) Use the product rule to differentiate y = (31:2 + (08\" d 3) Find the velocity v = df of an object falling through the atmosphere at a height 31 of y = 2000 + 15t 0.002t4 Give the velocity after t = 3 seconds 4) Using the chain rule, calculate the derivative 3: of the function y = In (bx3 7) . d. and so determine the value of 3% when x = 2. 5) A cantilevered beam is subjected to a complex periodic wind loading and experiences a vibration at its unsupported end of x = 0.63 + csin(3t) 0.25 cos(5t) metres dx Find an expression for the velocity v = E of the unsupported end and give the value of the velocity when t = 0.3 seconds. 6) Integrate a) f(5t3 + g)dt b) f0! e'm" + 4 cos(2x))dx 7) The area under the force-displacement (Rx) graph of a spring under compression represents the work done in compressing the spring. .9 Work done = f F dx 0 If the force F required to compress a rubber spring is given by F = 7.5x + 120x3 Newtons where x is the displacement from its resting position. Calculate the work done in compressing the spring from rest by a displacement of g metres HE04ilssue 6fJune 2022 8) In an AC signal processing circuit, the mean voltage of a half wave rectier over a time period T is given by me = 1% for 1: sin (100m) dt Calculate the mean voltage over a time period T of 0.0045. 9) In a spark ignition engine, a mixture of air and petrol vapour is compressed in a cylinder starting at a volume V of me = 0.04 m3 to a volume of Vend = 0.008 m3. The work done in moving the piston is given by 0. 008 Work done = - I P (it! where P is the Pressure 0.04 If the fuelair mixture behaves as a polytropic uid under compression and obeys the power law relationship P= 47Vk determine the work done by the piston. 10) When a bearing fails in a vertical boring machine, the rate of temperature rise is observed to be where the temperature T is measured in Kelvin and time c is measured in seconds The initial temperate of the hearing at the moment of failure was recorded as 298 K (25C). By using an appropriate integration method give the equation that describes the temperature T in terms of the time t