Calc 3 Review Problems, Study notes of Mathematics

Calc 3 General Review Exam 3. Not specific for any professor, is just a helpful review for calculus.

Typology: Study notes

2025/2026

Uploaded on 05/04/2026

manasi-thapa
manasi-thapa 🇺🇸

1 document

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Calculus 3: Exhaustive Final Review (Ch 14 & 16)
Section 14.7: Max/Min & Saddle Points
1. f(x, y) = x^2 + xy + y^2 + y
2. f(x, y) = xy 2x 2y x^2 y^2
3. f(x, y) = (x y)(1 xy)
4. f(x, y) = x^4 + y^4 4xy + 1
5. f(x, y) = e^x cos(y)
6. f(x, y) = y cos(x), on [0, 2] × [0, 2]
7. Find absolute max/min of f(x, y) = x^2 + y^2 2x on triangle with vertices (2,0), (0,2), (0,2).
8. Find absolute max/min of f(x, y) = 3 + xy x 2y on region bounded by y = x^2 and y = 4.
9. Find three positive numbers whose sum is 100 and maximize the product.
10. Find the volume of the largest rectangular box in the first octant with one vertex on x + 2y + 3z
= 6.
pf3
pf4
pf5

Partial preview of the text

Download Calc 3 Review Problems and more Study notes Mathematics in PDF only on Docsity!

Calculus 3: Exhaustive Final Review (Ch 14 & 16)

Section 14.7: Max/Min & Saddle Points

  1. f(x, y) = x^2 + xy + y^2 + y
  2. f(x, y) = xy  2x  2y  x^2  y^
  3. f(x, y) = (x  y)(1  xy)
  4. f(x, y) = x^4 + y^4  4xy + 1
  5. f(x, y) = e^x cos(y)
  6. f(x, y) = y cos(x), on [0, 2] × [0, 2 ]
  7. Find absolute max/min of f(x, y) = x^2 + y^2  2x on triangle with vertices (2,0), (0,2), (0,2).
  8. Find absolute max/min of f(x, y) = 3 + xy  x  2y on region bounded by y = x^2 and y = 4.
  9. Find three positive numbers whose sum is 100 and maximize the product.
  10. Find the volume of the largest rectangular box in the first octant with one vertex on x + 2y + 3z = 6.

Section 14.8: Lagrange Multipliers

  1. f(x, y) = 3x + y, subject to x^2 + y^2 = 10
  2. f(x, y) = xy, subject to 4x^2 + y^2 = 8
  3. f(x, y, z) = x + 2y + 3z, subject to x^2 + y^2 + z^2 = 25
  4. f(x, y, z) = x^2 + y^2 + z^2, subject to x + y + z = 1
  5. f(x, y, z) = xyz, subject to x^2 + 2y^2 + 3z^2 = 6
  6. Find points on z^2 = x^2 + y^2 closest to (4, 2, 0).
  7. Maximize f(x, y, z) = x + y + z subject to x^2 + y^2 = 1 and y + 2z = 1.
  8. Find extreme values of f(x, y) = e^(xy), subject to x^2 + y^2 = 16.

Section 16.2: Line Integrals

  1. _C y^3 ds, where C: x = t^3, y = t, 0  t  2.
  2. _C x y^4 ds, where C is right half of x^2 + y^2 = 16.

Section 16.4: Green's Theorem

  1. ■_C x y^2 dx + 2x^2 y dy, triangle region.
  2. ■_C (y + e^(x)) dx + (2x + cos(y^2)) dy.
  3. ■_C (3y  e^(sin x)) dx + (7x + (y^4 + 1)) dy.
  4. ■_C (y^2 cos x) dx + (2y sin x + e^y) dy.
  5. Use Green’s Theorem to find area of ellipse x^2/a^2 + y^2/b^2 = 1.

Section 16.7: Surface Integrals

  1. ■_S x^2 z dS over cone z^2 = x^2 + y^2.
  2. ■_S (x + y + z) dS over unit cube.
  3. ■_S F · dS, F = , paraboloid.
  4. ■_S F · dS, F = , sphere.
  5. Flux of F = across plane x + y + z = 1.

Section 16.8–16.9: Stokes & Divergence

  1. Stokes’ Theorem problem with F =.
  2. Show circulation is zero for F =.
  3. Evaluate ■ curl F · dS over hemisphere.
  4. Divergence Theorem: F = over unit sphere.
  5. Divergence Theorem: F =.
  6. Divergence Theorem: F =.
  7. Flux of F = over given region.