MATH127-Final Exam-Practice Problems & Solutions
Spring2025-Chapters13,14,15,16,andSections17.1-17.3
th
TimeandLocation:Wednesday,May14 ,2025,4:30pm-7:00pm,RoomTBA.
Review:Friday 5/9 1:00-3:00 pm,Budig 110
Thefollowingisalistofimportantconceptsfromthesectionswhichwere not coveredbyMidtermExam
1 or2.StudytheMidterm1and2reviewmaterialforchaptersthatarenotcoveredhere.Thisisnot
acompletelistofthematerialthatyoushouldknowforthesechapters,butitisagoodindicationof
whatwillbeemphasizedontheexam.Athoroughunderstandingofallofthefollowingconceptswill
helpyouperformwellontheexam.Someplacestofindproblemsonthesetopicsarethefollowing:in
thebook,intheslides,inthelabworksheets,onquizzes,andinAchieve.
Supposethat isapotentialfunctionfor .Findandsketch
⃗(x,y )= ∇ f (x,y )= ⟨ 2x, − 1⟩
F
or
1. f(x,y) = x2 − y F⃗ F⃗.
,2. Suppose f (x,y )= p x 2 + y 2 isapotentialfunctionfor ⃗.Findandsketch
F ⃗.
F
* x y +
⃗(x,y )= ∇ f (x,y )=
F , .
p x2 + y 2 p x 2 + y2
3. Calculatethecurlanddivergenceofthefollowingvectorfields:
⃗(x,y,z )= ⟨ xyz, 0, − x 2y ⟩ .
(A ) F
2
,
Spring2025-Chapters13,14,15,16,andSections17.1-17.3
th
TimeandLocation:Wednesday,May14 ,2025,4:30pm-7:00pm,RoomTBA.
Review:Friday 5/9 1:00-3:00 pm,Budig 110
Thefollowingisalistofimportantconceptsfromthesectionswhichwere not coveredbyMidtermExam
1 or2.StudytheMidterm1and2reviewmaterialforchaptersthatarenotcoveredhere.Thisisnot
acompletelistofthematerialthatyoushouldknowforthesechapters,butitisagoodindicationof
whatwillbeemphasizedontheexam.Athoroughunderstandingofallofthefollowingconceptswill
helpyouperformwellontheexam.Someplacestofindproblemsonthesetopicsarethefollowing:in
thebook,intheslides,inthelabworksheets,onquizzes,andinAchieve.
Supposethat isapotentialfunctionfor .Findandsketch
⃗(x,y )= ∇ f (x,y )= ⟨ 2x, − 1⟩
F
or
1. f(x,y) = x2 − y F⃗ F⃗.
,2. Suppose f (x,y )= p x 2 + y 2 isapotentialfunctionfor ⃗.Findandsketch
F ⃗.
F
* x y +
⃗(x,y )= ∇ f (x,y )=
F , .
p x2 + y 2 p x 2 + y2
3. Calculatethecurlanddivergenceofthefollowingvectorfields:
⃗(x,y,z )= ⟨ xyz, 0, − x 2y ⟩ .
(A ) F
2
,
