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Examination I - Fall 2008 - Calculus III | MATH 2210, Exams of Advanced Calculus

University of Utah (The U)Advanced Calculus
Prof. Robert Palais

Material Type: Exam; Professor: Palais; Class: Calculus III; Subject: Mathematics; University: University of Utah; Term: Fall 2008;

Typology: Exams

Pre 2010

Uploaded on 08/30/2009

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Calculus III, Math 2210-90, Fall 2008, Bob Palais
Exam 1, Chapter 11
Vectors, Curves, and Surfaces
Show all your work on the exam for full credit. You may use graphing (or regular)
calculators.
The main topics covered in this exam are:
1. Vector geometry: The dot product of vectors and its geometric interpretation in
terms of length and angle. Unit vectors. The orthogonal projection of one vector on
another. The cross product of vectors and its geometric interpretation in terms of length
and angle, and in terms of area.
2. Vector equations of lines in 2D and 3D in various forms (e.g. parametric, symmet-
ric) and from various data (e.g. two points, point and parallel vectors)
3. Vector equations of planes in 3D in various forms (e.g. parametric, point-normal)
and from various data (e.g. three points, point and parallel vectors)
4. Distance from a point to a line or plane. Distance between two non-intersecting
lines or planes.
5. Tangent lines of curves
6. Velocity and acceleration in 3D, tangential and normal components of acceleration,
curvature.
1
pf3

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Download Examination I - Fall 2008 - Calculus III | MATH 2210 and more Exams Advanced Calculus in PDF only on Docsity!

Calculus III, Math 2210-90, Fall 2008, Bob Palais Exam 1, Chapter 11 Vectors, Curves, and Surfaces

calculators.^ Show all your work on the exam for full credit.^ You may use graphing (or regular) The main topics covered in this exam are:

  1. Vector geometry: The dot product of vectors and its geometric interpretation in terms of length and angle.another. The cross product of vectors and its geometric interpretation in terms of length Unit vectors. The orthogonal projection of one vector on and angle, and in terms of area.

ric) and from various data (e.g. two points, point and parallel vectors)2. Vector equations of lines in 2D and 3D in various forms (e.g. parametric, symmet- and from various data (e.g. three points, point and parallel vectors)3. Vector equations of planes in 3D in various forms (e.g. parametric, point-normal)

lines or planes.4.^ Distance from a point to a line or plane.^ Distance between two non-intersecting

  1. Tangent lines of curves
  2. Velocity and acceleration in 3D, tangential and normal components of acceleration, curvature.
  1. Let W = 1I + 1J + 1K and N = − 2 I + 2J + 1K. a) (10 points) Write W as the sum of a vector parallel to N and a vector orthogonal to N. b) (10 points) Find the distance from the point (1, 1 , 1) to the plane − 2 x + 2y + 1z = 0. (Hint: Use part a.)
  2. (20 points) Let P = (1, 0 , 0), Q = (2, 0 , −1), R = (2, 2 , 0). a) (10 points) Find a unit vector normal to the plane containing P, Q, and R. b) (10 points) Find the parametric form of the line containing the point P parallel to the vector P Q~. Also find the symmetric equations for this line.
  3. (20 points) Let U and V be vectors in space satisfying:

U · U = V · V = 1, U · V =^12 ,

and U × V =

2 I^ + 0^ J^ + 0^ K.

Find the following quantities: a) (5 points) ||U + V||^2 = b) (5 points) The angle θ between U and V: c) (5 points) (V − U) · (U × V) = d) (5 points) V × (2U + 3V) =

  1. Let C be the curve in the space determined by R(t) = cos(t) I + sin(t) J + t K for 0 ≤ t ≤ 2 π. a) (10 points) Find the parametric equations for the line tangent to C at t = π. b) (10 points) Find the length of C.