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09:46 4G Done Unit 7 B assignment (1 of 2) Q ALWAYS LEARNING PEARSON a) Velocity v(ms-1) - given v(0) = 10ms-1 b) Displacement s(m) - given s(0) = 5m c) Calculate the values of v and s for: i) t = 2s ii) t = 5s 3 The extension, y, of a material with an applied force, F, is given by y = efx1x10-3 a) Calculate the work done if the force increases from 100N to 500N using: i) An analytical integration technique ii) A numerical integration technique [Note: the work done is given by the area under the curve] b) Compare the two answers c) Using a computer spreadsheet increase the number of values used for your numerical method d) Analyse any effect the size of numerical step has on the result. 4 An electronic signal can be modelled by the function v = 12 sin 40. Calculate the: a) Mean b) Root mean square (RMS) Over a range of 0 s 0 S - radians. [Note the trigonometric identity cos 20 = 1 - 2 sin2 0] 5 A complex function can be modelled by the equation: y = cos(x3) 3x2 Find the indefinite integral of the function (f cos(x3) 3x2 dx) using a substitution method. 6 The acceleration of a particle moving in a strange way has been modelled as a = ex. a) Use integration by parts to find an equation to model the velocity v if v = f exx dx. b) Is the problem any different if you find v = ] xe* dx? 2 BTEC Assignment Brief v1.0 BTEC Internal Assessment QDAM January 2015 :E\f09:46 4G Done Unit 7 B assignment (1 of 2) Q A 3 of 4 ALWAYS LEARNING PEARSON 7 Newton's laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation: dt dl = -k(T - Ta) This can also be written as: dT T - Ta - = -k(T - Ta) Where: T = Temperature of material Ta = Ambient (room) temperature k = A cooling constant a) Integrate both sides of the equation and show that the temperature difference is given by: (T - Ta) = Coe-kt [C. is a constant for this problem] b) Calculate C. if the initial temperature is 70 C and To = 20 C. Checklist of evidence Your informal report should contain: required analysis worked solutions to the problems Each worked solution should be laid out clearly and contain brief explanations of the stages of the calculation to indicate your understanding of how calculus can be used to solve an engineering problem. Graphs should be well presented and clearly labelled and comparisons between methods should be accurate and well presented. Criteria covered by this task: Unit/Criteria reference To achieve the criteria you must show that you are able to: Evaluate, using technically correct language and a logical structure, the 7/B.D1 correct integral calculus and numerical integration solutions for each type of given routine and non-routine functions, including at least two set in an engineering context. Find accurately the integral calculus and numerical integration solutions 7/B.M2 for each type of given routine and non-routine function, and find the properties of periodic functions. 7/B.P4 Find the indefinite integral for each type of given routine function. 3 BTEC Assignment Brief v1.0 BTEC Internal Assessment QDAM January 2015 :E09:46 4G Done Unit 7 B assignment (1 of 2) Q ALWAYS LEARNING PEARSON BTEC BTEC Assignment Brief Pearson BTEC Level 3 National Extended Diploma in Engineering Pearson BTEC Level 3 National Extended Diploma in Electrical/Electronic Engineering Pearson BTEC Level 3 National Extended Diploma in Qualification Mechanical Engineering Pearson BTEC Level 3 National Extended Diploma in Computer Engineering Pearson BTEC Level 3 National Extended Diploma in Manufacturing Engineering Pearson BTEC Level 3 National Extended Diploma in Aeronautical Engineering Unit number and title Unit 7: Calculus to solve engineering problems Learning aim(s) (For NQF B: Examine how Integral calculus can be used to solve only engineering problems Assignment title Solving engineering problems that involve differentiation Assessor Issue date Hand in deadline You are working as an apprentice engineer at a company involved in the research, design production and maintenance of bespoke engineering solutions for larger customers. Vocational Scenario or Context Part of your apprenticeship is to spend time working in all departments, however a certain level of understanding needs to be shown before the managing director allows apprentices into the design team and so she has developed a series of questions on integration to determine if you are suitable. 1 The tasks are to: a) Find the indefinite integral of the function y = 3t2 + 2e3t +=+ 2 cos 3t Task 1 b) Calculate the definite integral [" 312 + 2e31 + = + 2 cos 3t dt 2 A vehicle is moving with a uniform acceleration a = 2ms-2, determine the functions for: :E