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Problems of Calculus of Several Variables - Final Exam | MATH 231, Exams of Mathematics

Pennsylvania State University - AbingtonMathematics

Material Type: Exam; Class: Calculus of Several Variables; Subject: Mathematics; University: Penn State - Main Campus; Term: Fall 2012;

Typology: Exams

2011/2012

Uploaded on 12/20/2012

dan-givigliano
dan-givigliano ๐Ÿ‡บ๐Ÿ‡ธ

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Math 231 Practice Final Exam Problems
Note: This is a selection of problems that you should try to solve in a testlike setting to see if you
have studied enough. The actual exam has 12 problems, while this one has 16, so it may take longer
than the exam. There will be no printend solutions, but you should come to office hours or ask
your classmates if you cannot figure out an answer.
1. The equation of the line that contains the point (1,โˆ’1,3) and is orthogonal to the plane
2x+ 3yโˆ’z= 1 is given by
(a) x= 1 + 2t, y =โˆ’1+3t, z = 3 โˆ’t
(b) x= 1 + t, y =โˆ’1+4t, z = 3 โˆ’3t
(c) x= 2 + t, y = 3 + 4t, z =โˆ’1โˆ’3t
(d) x= 1 + 2t, y = 4 โˆ’3t, z =โˆ’3โˆ’t
2. Suppose aยทb= 6โˆš2, |a|= 4, |b|= 3. Then |aร—b|=
(a) 6
(b) 6โˆš2
(c) 2โˆš3/3
(d) 3โˆš2
3. At what point (x, y) is the tangent plane of f(x, y) = 2x2+y2parallel to the plane xโˆ’y+ 2z=
1?
(a) (1/8,โˆ’1/4)
(b) (โˆ’1/2,1)
(c) (1/2,โˆ’1)
(d) (โˆ’1/8,1/4)
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Download Problems of Calculus of Several Variables - Final Exam | MATH 231 and more Exams Mathematics in PDF only on Docsity!

Math 231 Practice Final Exam Problems

Note: This is a selection of problems that you should try to solve in a testlike setting to see if you have studied enough. The actual exam has 12 problems, while this one has 16, so it may take longer than the exam. There will be no printend solutions, but you should come to office hours or ask your classmates if you cannot figure out an answer.

  1. The equation of the line that contains the point (1, โˆ’ 1 , 3) and is orthogonal to the plane 2 x + 3y โˆ’ z = 1 is given by (a) x = 1 + 2t, y = โˆ’1 + 3t, z = 3 โˆ’ t (b) x = 1 + t, y = โˆ’1 + 4t, z = 3 โˆ’ 3 t (c) x = 2 + t, y = 3 + 4t, z = โˆ’ 1 โˆ’ 3 t (d) x = 1 + 2t, y = 4 โˆ’ 3 t, z = โˆ’ 3 โˆ’ t
  2. Suppose a ยท b = 6

2, |a| = 4, |b| = 3. Then |a ร— b| = (a) 6 (b) 6

(c) 2

(d) 3

  1. At what point (x, y) is the tangent plane of f (x, y) = 2x^2 +y^2 parallel to the plane xโˆ’y +2z = 1? (a) (1/ 8 , โˆ’ 1 /4) (b) (โˆ’ 1 / 2 , 1) (c) (1/ 2 , โˆ’1) (d) (โˆ’ 1 / 8 , 1 /4)
  1. Let 4 be the triangle determined by the points (1, 0 , 1), (2, 2 , 2), and (โˆ’ 1 , 0 , 0).

(a) Find the equation of the plane containing 4. (b) Find the area of 4.

  1. Let f (x, y) = y^2 cos(x).

(a) Is f (x, y) increasing or decreasing in the direction of โˆ’ 3 i + 4j at (0, โˆ’2)? (b) Find the maximum rate of change of f (x, y) at (0, โˆ’2). (c) In what direction does the maximum rate of change occur?

  1. Let C be the curve given by r(t) = i โˆ’ t^2 j + (1 + t^2 )k. (a) Find parametric equations of the tangent line to C at the point (1, โˆ’ 1 , 2). (b) Calculate the arclength of r(t) from t = 0 to t = 1.
  2. Find all second partial derivatives of f (x, y, z) = xy/z^ at (x, y, z) = (2, 3 , 4).
  3. (a) Find a parametric equation for the line of intersection of the planes x โˆ’ z = 1 and y + 2z = 3 (b) Write an equation for a plane which is orthogonal to both of these planes.
  4. (a) Sketch the surface x^2 โˆ’ y^2 + z^2 โˆ’ 2 y โˆ’ 2 = 0. (b) Add to your sketch the curve of intersection with the plane y = 0 and write a parametriza- tion r(t) = ใ€ˆx(t), y(t), z(t)ใ€‰ for this curve.
  1. Define the function f (x, y) = x^2 y โˆ’ x^2 โˆ’ y^2 โˆ’ 2 y + 2. Find the critical points of f (x, y) and classify them as minima, maxima, and saddle points.
  2. Find the volume of the largest rectangular box in the first octant which with three faces in the coordinate planes and one corner in the plane x + 2y + 3z = 6.
  3. Find the maximum value of f (x, y) = x + y โˆ’ xy, on the domain bounded by the triangle with vertices (2, 0), (0, 2), (0, โˆ’2).
  4. Let f (x, y) = x^4 โˆ’ y^4. Using Lagrange multipliers, find the maximum value and the minimum value of f (x, y) subject to the constraint x^2 + y^2 = 1. (If you do not use Lagrange multipliers, you will not receive credit).