Department of MathematicsAnna UniversityMA7151 Mathematics – I (Regulation 2015)Differential Calculus (Tutorial Problems-2016)1.A open box with an open top is to be constructed from a rectangular piece of cardboard withdimension 12 cm by 20 cm by cutting out equal square of sidexat each corner and then folding upsides as in the figure. Express the volumeVof the box as the function ofx2.A spherical balloon with radiusrinches has volume34)(rrV. Find a function that represents theamount of air required to inflate the balloon from a radius ofrinches to a radius of1rinches.3.A stone is dropped into a lake, creating ripple that travels outward at a speed of 60cm/s.(a) Express the radiusrof this circle as a function of the timet(in seconds).(b) IfVis the volume of the balloon as a function of the radius, findrVand interpret it.4.A bacterial culture stats with 500 bacteria and doubles in size every half hour.(a)How many bacteria are there after 3 hours?(b)How many bacterial are there afterthour?(c)How many bacterial are there after 40 minutes?5.The graph of the functionfis shown in the following figure(i)Find the value of)1((ii)Estimate the value of)2((iii)For what value ofxis2)(x(iv)State the domain and range of(v)On what interval isfincreasing?(vi)On what interval isfdecreasing6.Ifxxxf2)(andtttg2)(, is it true thatgf7.If11)(xxxfand1)(xxg, is it true thatgf8.Sketch the graph of the following functions and hence find its domain and range:(i)12)(xxf,(ii)1)3()(2xxf,(iii)0,10,2)(xifxxifxxf,(iv)xxftan)((v)xxfsin1)(, (vi)xxf21)(, (vii)3)(xxf,(viii)xxf2cos)9.Define the absolute value functionxxf)((i)Sketch the function(ii)Find its domain and range.(iii) Is it continuous?(iv) Is it differentiable?10.A functionfis given by1,1,1)(2xifxxifxxf. Evaluate the values)2(f,)1(f,)0(fand)4(.3ffff?2?(.,f.