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MATH 2500 Final Practice Problems: Fall 2024, Summaries of Mathematics

University of Georgia (UGA)Mathematics

A set of practice problems for math 2500, covering various topics including vector operations, lines and planes in 3d space, multivariable calculus, and line integrals. Each problem is categorized by its corresponding learning outcome, allowing students to focus on specific areas of weakness. A valuable resource for students preparing for the final exam.

Typology: Summaries

2023/2024

Uploaded on 12/10/2024

gage-kelley
gage-kelley 🇺🇸

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MATH 2500 Final PRACTICE
Fall 2024
No. LO Score Notes/Suggestions No. LO Score Notes/Suggestions
1 2B 6 4H
2 2D 7 5B
3 2G 8 5E
4 4C 9 6B
5 4F 10 6D
1. (Outcome 2B) For this problem, consider the vectors v=⟨2,−4,1⟩and w=⟨−2,1,3⟩.
(a) Determine v·w.
(b) Determine cos(θ), where θis the angle between vand w. (Note that I did NOT ask you
to solve for θ.)
(c) Determine projwv, the vector projection of vonto w.
(d) Is vorthogonal to w? Why or why not?
pf3
pf4
pf5
pf8
pf9

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Download MATH 2500 Final Practice Problems: Fall 2024 and more Summaries Mathematics in PDF only on Docsity!

MATH 2500 Final PRACTICE Fall 2024

No. LO Score Notes/Suggestions No. LO Score Notes/Suggestions 1 2B 6 4H

2 2D 7 5B

3 2G 8 5E

4 4C 9 6B

5 4F 10 6D

  1. (Outcome 2B) For this problem, consider the vectors v = ⟨ 2 , − 4 , 1 ⟩ and w = ⟨− 2 , 1 , 3 ⟩.

(a) Determine v · w.

(b) Determine cos(θ), where θ is the angle between v and w. (Note that I did NOT ask you to solve for θ.)

(c) Determine projwv, the vector projection of v onto w.

(d) Is v orthogonal to w? Why or why not?

  1. (Outcome 2D) For this problem, consider the points P (0, 3 , −2) and Q(5, 1 , 0) and vectors v = ⟨ 2 , − 4 , 1 ⟩ and w = ⟨− 2 , 1 , 3 ⟩.

(a) Determine a vector equation of the line through P with direction vector v.

(b) Determine a vector equation of the line segment from P to Q.

(c) Determine a scalar equation of the plane that contains P and has normal vector v.

(d) Determine a vector equation of the line through P that is orthogonal to the plane − 4 x + y − 5 z + 12 = 0.

(e) Determine a scalar equation of the plane through P that contains v and w.

  1. (Outcome 4C) Determine ∂f ∂x and ∂f ∂y for f (x, y) = 3x^3 − 2 y^2 + 6xy − 4 y + 8x^5 y − 1.
  2. (Outcome 4F) Consider the function f (x, y, z) = 4yz − xy^2 + 8z + 1.

(a) Determine the gradient function ∇f of f.

(b) Determine the directional derivative of f at the point (0, 1 , −1) in the direction of the vector v = ⟨− 1 , 1 , 2 ⟩.

  1. (Outcome 4H) Use the second derivatives test to determine all local extrema and saddle points of the function f (x, y) = 3xy − 2 x^2 − 2 y^2 + 12x − 4 y.
  1. (Outcome 5E) Evaluate the integral

y E 8 x^2 z dV where E is the solid bounded above by the elliptical paraboloid z = 2 − x^2 − y^2 and bounded below by the xy-plane.

  1. (Outcome 6B)

(a) Use the cross-partials property or the curl criterion to show that the vector field F = ⟨ 2 xy + 2yz, 2 xz − 4 z, 2 xy − 4 y⟩ is not conservative.

(b) Show that the vector field F = ⟨ 6 xy^2 − 3 y, 6 x^2 y − 3 x + 2y − z, −y⟩ is conservative by finding a potential function for F.