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KMA252

Calculus&Applications2

Assignment1. Due9amMondayMarch24,2025

IntendedLearningOutcomes

ILO1 : Interrogatethebehaviourofmultivariablefunctionsusingavarietyofanalyticaltechniques.

ILO6 : DevelopanduseFourierSeriestechniquesforperiodicfunctions.

SpecificLearningOutcomes: Thisassignmentwillgiveyoupracticein

• SolvingforthecoefficientsofaFourierseries.

• ConstructingtheFourierseriesforagivenperiodicfunction.

• Computing, drawing, and interpreting cross-sections and contours of a three dimensional

surface.

• Reformingtheequationofasurfaceintostandardform.

• Parameterisingaspacecurveandasurface.

Forrelatedtopicsandsimilarexamplesandproblems,refertoThomas†Sections19.1

PeriodicFunctions,19.2SummingSinesandCosines,19.4Approximationsof

Functions,6.1VolumesUsingCross-Sections(fordefinitionofcross-section),11.6

ConicSections,11.1Three-DimensionalCoordinateSystems,11.6Cylindersand

QuadricSurfaces,and13.1FunctionsofSeveralVariables.

Week1Questions

1. Considerthepiecewisefunction



 −2, −2

f(x)=

 x2−2, 0

Thefunctionistobeperiodicallyextendedto±∞.

(i) Drawthefunctionanditsextensionontheinterval[−6,10].

(ii) Does f exhibit even or odd behaviour? What is the consequence for the Fourier series - which

termswillbeincluded?

(iii) FindtheFouriercoefficienta .

0

(iv) FindthesimplifiedgeneralexpressionfortheFouriercoefficientsa , n>0.

n

(v) FindthesimplifiedgeneralexpressionfortheFouriercoefficientsb , n>0.

n

(vi) WhatistheFourierseriesF(x)off(x)?

(vii) WhatvaluewilltheFourierseriescomputetoatx=70?

†Thomas’Calculus,15thEditioninSIUnits.Haasetal.Pearson2024.

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(viii) WhatistheoverallconvergencerateoftheFourierseries?

Week2Questions

3. Considerthefunction f(x,y)=2x2+4x+my2+3y+2 where m∈R.

(i) Modifyf usingcompletionofsquaresonxandy.

(ii) Considerthesurface z = f(x,y) andcontours z = c, c ∈ R. Forthefollowingcases, putthe

equationinstandardform,describethestandardform,anddeterminetheconditionsonc.

(A) m>0,

(B) m<0,

(C) m=0.

(iii) Parametricallydefinethecontoursform>0.

Week3Questions

3. Figure3showsaplaneandatwo-sheetedsurface,givenrespectivelyby

S ={x,y,z ∈R3|y−2x=16} and

1

S ={x,y,z ∈R3|x2−6y2−2z2−2x+24y−4z =41}.

2

Aspacecurve C,tracingoutthelineofintersectionofthetwosurfaces,isdepictedbythebluecurve.

Figure3

(i) Rewrite S in standard form then describe the surface, giving its name, geometric centre, and

2

orientation.

(ii) ParametricallydefinethesheetofS forwhichx<0.

2

(iii) Determineparametricdefinitionsfor x,y, andz todescribethelineofintersection C.

†Thomas’Calculus,15thEditioninSIUnits.Haasetal.Pearson2024.