AM10CO2020.dvi AM10CO 1. Consider the function f(x) = x2 x3 + 4x . (a) Determine the domain and parity (if any) of the function. (3 marks) (b) Determine the critical points (maxima, minima and...

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AM10CO2020.dvi AM10CO 1. Consider the function f(x) = x2 x3 + 4x . (a) Determine the domain and parity (if any) of the function. (3 marks) (b) Determine the critical points (maxima, minima and inflection points) of f(x). (6 marks) (c) Determine regions of the domain where f(x) is increasing, decreasing, concave and convex. (6 marks) (d) Determine the roots of f(x). (3 marks) (e) Plot the function f(x) using the provided graph paper. (12 marks) Total: 30 Marks 1 of 3 This Assessment is Subject to the University Assessment Regulations for Candidates AM10CO 2. Consider the following differential equation: dy(x) dx + 2xy(x) = f(x) , (1) where f(x) =    0 x < 0,="" −2x3="" 0="">< x="">< 1,="" 0="" 1="">< x.="" (2)="" (a)="" write="" and="" solve="" the="" associated="" homogeneous="" equation="" of="" (1).="" (4="" marks)="" (b)="" propose="" a="" solution="" to="" (1),="" for="" each="" of="" the="" regions="" where="" the="" non="" homoge-="" neous="" term="" (2)="" has="" been="" defined,="" based="" on="" your="" results="" from="" (a)="" above.="" (6="" marks)="" (c)="" find="" a="" solution="" to="" the="" full="" problem.="" (14="" marks)="" (d)="" find="" a="" continuous="" solution="" to="" the="" full="" problem.="" (6="" marks)="" (e)="" adjust="" the="" free="" constant="" according="" to="" the="" condition="" y(1)="1." (5="" marks)="" total:="" 35="" marks="" 2="" of="" 3="" this="" assessment="" is="" subject="" to="" the="" university="" assessment="" regulations="" for="" candidates="" am10co="" 3.="" consider="" the="" the="" following="" second-order="" differential="" equation:="" d2y(x)="" dx2="" −="" 2="" dy(x)="" dx="" +="" 2y(x)="2[sin(x/2)]2." (3)="" (a)="" write="" and="" solve="" the="" associated="" homogeneous="" equation="" of="" (3).="" (10="" marks)="" (b)="" propose="" a="" solution="" to="" (3),="" based="" on="" your="" results="" from="" (a)="" above.="" (4="" marks)="" (c)="" find="" the="" solution="" to="" the="" full="" problem.="" hint:="" remember="" that="" 2[sin(x/2)]2="1−" cos(x).="" (16="" marks)="" (d)="" adjust="" the="" free="" constants="" according="" to="" the="" conditions="" y(0)="1" and="" y′(0)="0." (5="" marks)="" total:="" 35="" marks="" end="" of="" assessment="" 3="" of="" 3="" this="" assessment="" is="" subject="" to="" the="" university="" assessment="" regulations="" for="" candidates="" am10co2019.dvi="" am10co="" 1.="" consider="" the="" function="" f(x)="{" 1="" x="0" sin(x)="" x="" otherwise="" .="" (a)="" determine="" the="" domain="" and="" parity="" (if="" any)="" of="" the="" function.="" (3="" marks)="" (b)="" determine="" the="" expressions="" satisfied="" by="" the="" critical="" points="" (maxima,="" min-="" ima="" and="" inflection="" points),="" solving="" them="" for="" specific="" values="" when="" possible="" (12="" marks)="" (c)="" determine="" the="" roots="" of="" f(x).="" (3="" marks)="" (d)="" plot="" the="" function="" f(x)="" using="" the="" provided="" graph="" paper.="" (12="" marks)="" total:="" 30="" marks="" 1="" of="" 3="" this="" examination="" is="" subject="" to="" the="" examination="" regulations="" for="" candidates="" gilfoylh="" typewritten="" text="" gilfoylh="" typewritten="" text="" .="" gilfoylh="" typewritten="" text="" gilfoylh="" typewritten="" text="" am10co="" 2.="" find="" the="" continuous="" solution="" of="" the="" following="" equation:="" 3(x2="" +="" 7)="" dy(x)="" dx="" +="" 2xy(x)="f(x)" ,="" where="" f(x)="" ="" ="" 0="" x="">< 0="" x="" 0="">< x="">< 1="" 0="" 1="">< x , satisfying the condtiton y(1) = 0. total: 35 marks 2 of 3 this examination is subject to the examination regulations for candidates gilfoylh highlight typo am10co 3. solve the following equation: d2y(x) dx2 + dy(x) dx − 2y(x) = x+ sin(x) with y(0) = 1 and y′(0) = 0. total: 35 marks end of examination paper 3 of 3 this examination is subject to the examination regulations for candidates x="" ,="" satisfying="" the="" condtiton="" y(1)="0." total:="" 35="" marks="" 2="" of="" 3="" this="" examination="" is="" subject="" to="" the="" examination="" regulations="" for="" candidates="" gilfoylh="" highlight="" typo="" am10co="" 3.="" solve="" the="" following="" equation:="" d2y(x)="" dx2="" +="" dy(x)="" dx="" −="" 2y(x)="x+" sin(x)="" with="" y(0)="1" and="" y′(0)="0." total:="" 35="" marks="" end="" of="" examination="" paper="" 3="" of="" 3="" this="" examination="" is="" subject="" to="" the="" examination="" regulations="" for="">