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Calculus in 3D Practice Final Exam, Summaries of Calculus

Dartmouth CollegeCalculus

CARNEGIE MELLON UNIVERSITY. Math 21-259 Calculus in 3D. Practice Final Exam. Allowed Time: 150 mins. 1. (15 points) Find symmetric equations for the line of ...

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2022/2023

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DEPARTMENT OF MATHEMATICAL SCIENCES
CARNEGIE MELLON UNIVERSITY
Math 21-259 Calculus in 3D
Practice Final Exam
Allowed Time: 150 mins
1. (15 points) Find symmetric equations for the line of intersection Lof the two planes
x+y+z= 1 and x−2y+ 3z= 1.Also, find the angle between these two planes.
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DEPARTMENT OF MATHEMATICAL SCIENCES

CARNEGIE MELLON UNIVERSITY

Math 21-259 Calculus in 3D Practice Final Exam

Allowed Time: 150 mins

  1. (15 points) Find symmetric equations for the line of intersection L of the two planes x + y + z = 1 and x − 2 y + 3z = 1. Also, find the angle between these two planes.
  1. (15 points) Let r(t) = (

2 t, et, e−t).

(a) Calculate the arc length function s(t) measured from t = 0.

(b) Find the equation of the line tangent to the curve at the point r(1).

(c) Compute the unit tangent vector Tˆ(t).

(d) Compute κ(t).

  1. (20 points) Find the absolute maximum and minimum values of f (x, y) = x^2 + y^2 + x^2 y + 4 on the set D = {(x, y) : − 1 ≤ x ≤ 1 , − 1 ≤ y ≤ 1 }. Also, give the points at which the function attains its maximum and minimum values.
  1. (20 points) Find the dimensions of the rectangular box of maximum volume if the total surface area is given as 64 cm^2 using the method of Lagrange multipliers.
  1. (15 points) Find the volume of the solid in the first octant which is bounded by the cone x^2 + y^2 = 3z^2 , by the planes x = 0 and x =

3 y, and by the sphere 4x^2 + 4y^2 + 4z^2 = 1.

  1. (20 points) Evaluate

R(x^ +^ y) cos^ π(2x^2 +^ xy^ −^ y^2 ) dx^ dy^ where R is the parallelogram with vertices (0, 0), (1, -1), (1/3, 2/3), and (4/3, -1/3).

  1. (20 points) Compute the line integral of the vector field F(x, y) =< xy, x^2 y > over the boundary of the triangle with vertices (0, 0), (0, 1), (2, 1) directly and by using Green’s theorem.
  1. (10 points) Find the surface area of the part of the surface z = 1 + 3x + 2y^2 that lies above the triangle with vertices (0, 0), (0, 1), and (2, 1).