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MATH 264 review 2011, Exams of Mathematics

Abertay UniversityMathematics

/MATH 264 /review /MATH 264 /review /MATH 264 /review

Typology: Exams

2011/2012

Available from 01/08/2025

yacine-ouali
yacine-ouali 🇬🇧

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MATH 264 — R E V I E W E X A M 1
TOPICS COVERED (Chapters 13,14,15.1)
1. Vectors
add, subtract, multiply vectors, also graphically
magnitude of vectors
find resultant force if several forces are applied
2. Dot Product and Cross Product
compute dot and cross product of two vectors
magnitude and direction of cross product
what does a·b= 0 mean? what does a×b= 0 mean?
what is a·a? what is a×a?
know relation between dot/cross product and angle θ
find projections and orthogonal projections of bonto a
applications: compute areas of triangles, parallelograms, find components of force in
one direction, find vector normal to plane, etc
3. Lines and Planes
find equations for lines/planes
find intersections of lines and lines, lines and planes, planes and planes
find angle between lines and lines, lines and planes, planes and planes
determine when lines are parallel, skew, intersect
find distance from point to plane/line (dont memorize and apply formula)
4. Graphing in 3D
Graph any of the quadric surfaces in homework
Graph equations in cylindrical and spherical coordinates
5. Vector functions
Sketch elementary curves r(t) = hx(t), y(t), z(t)i(circles, helices, ellipses, lines,
curves which can be rewritten in nonparametric form)
derivative r′(t)and tangent vector
differentiation rules
velocity, speed, acceleration
find arclength
find particle position given acceleration (integrate)
6. Functions of several variables
Read tables representing functions of two variables
Find domains of functions of two and three variables
Represent functions z=f(x, y) of two variables by
Graphing the surface z=f(x, y)in x−y−zspace
Graphing contour (level) curves f(x, y) = const in the x−yplane.
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MATH 264 — R E V I E W E X A M 1

TOPICS COVERED (Chapters 13,14,15.1)

  1. Vectors add, subtract, multiply vectors, also graphically magnitude of vectors find resultant force if several forces are applied
  2. Dot Product and Cross Product compute dot and cross product of two vectors magnitude and direction of cross product what does a · b = 0 mean? what does a × b = 0 mean? what is a · a? what is a × a? know relation between dot/cross product and angle θ find projections and orthogonal projections of b onto a applications: compute areas of triangles, parallelograms, find components of force in one direction, find vector normal to plane, etc
  3. Lines and Planes find equations for lines/planes find intersections of lines and lines, lines and planes, planes and planes find angle between lines and lines, lines and planes, planes and planes determine when lines are parallel, skew, intersect find distance from point to plane/line (dont memorize and apply formula)
  4. Graphing in 3D Graph any of the quadric surfaces in homework Graph equations in cylindrical and spherical coordinates
  5. Vector functions Sketch elementary curves r(t) = 〈x(t), y(t), z(t)〉 (circles, helices, ellipses, lines, curves which can be rewritten in nonparametric form) derivative r′(t) and tangent vector differentiation rules velocity, speed, acceleration find arclength find particle position given acceleration (integrate)
  6. Functions of several variables Read tables representing functions of two variables Find domains of functions of two and three variables Represent functions z = f (x, y) of two variables by Graphing the surface z = f (x, y) in x − y − z space Graphing contour (level) curves f (x, y) = const in the x − y plane.

STUDY PROBLEMS

Here are some suggested study problems. Note that not all the topics are fully covered. You need to also go through the list of topics and do problems from the homework in those areas that are not extensively covered or that you feel a little weak in.

Vectors, Dot Product, Cross Product

  1. Express w in terms of the vectors u and v in the figure.

v

u

w

  1. Consider a quadrilateral with vertices A,B,C,D, as shown. Let a 1 = AB, a 2 = BC, a 3 = CD, a 4 = DA.

B

D C

A

(a) Graph the vectors b 1 = a 1 + a 2 , b 2 = a 1 + a 2 + a 3 , b 3 = a 1 + a 2 + a 3 + a 4 , (b) Show that a 1 + a 2 + a 3 + a 4 = 0 using the definition of a 1 ,... , a 4.

  1. §13.2, Exercise 4.
  2. Let F 1 , F 2 be as shown in Figure, where |F 1 | = 3 and |F 2 | = 4. (a) Graph the resultant force F in the figure. (b) Find the magnitude and elevation angle of the resultant force F. (Ans: |F|=

25+6(√ 6 −√2), θ=76. 24 o)

60 o 45 o

F 1 F 2

  1. (Chain problem) §13.2, Exercise 34 (Ans: 30. 09 N )
  2. If two vectors enclose an acute angle, they point in the same general direction. If they enclose an obtuse angle, they point in generally opposite directions. Are the following pairs of vectors perpendicular to each other? If not, do they point in the same general direction or in generally opposite directions? (a) a = 〈 1 , 2 〉, b = 〈− 2 , 4 〉 (b) a = 〈 0 , 2 , − 2 〉, b = 〈 1 , 0 , 3 〉 (c) a = 〈 2 , − 1 , 4 〉, b = 〈 2 , 0 , − 1 〉 (d) a = 〈 3 , 4 , 5 〉, b = 〈 2 , − 1 , 1 〉
  3. The following figures shows an equilateral triangle and a square, where |a| = 1. Find a · b and a · c, and a × b and c × b (both their magnitude and direction).

a (^) c

b

a

c

b

  1. (a) Derive the vector and scalar equations of the straight line through (xo, yo, zo) in the direction of the vector v = 〈a, b, c〉. Make a sketch using vectors. (b) Derive the vector and scalar equations of the plane containing the point (xo, yo, zo) and normal to the vector n = 〈a, b, c〉 Make a sketch using vectors.

Graphing in 3D

  1. Chapter 13 Review, Exercises: 1,26-

Vector functions

  1. Chapter 14 Review, Exercises: 1,4,5,6,8,9,16,17,18, (Ans to 4: r(t)=〈 1 , 1 , 1 ,〉+t〈 2 , 4 , 3 〉) (Ans to 6: (a)〈 158 , 0 ,− ln 2 〉, (b)r(t)=〈 1 , 1 , 0 ,〉+t〈− 3 , 2 , 1 〉, (c)−3(x−1)+2(y−1)+z=0) (Ans to 8: 272 (13^3 /^2 −8))
  2. The curvature of a curve r(t) is defined as κ = |dT/ds|, where T is the unit tangent vector to the curve and s is arclength. Curves are not typically parametrized with respect to arclength, making this formula difficult to apply. Instead, they are parametrized with respect to an arbitrary parameter t. Derive an alternative formula for κ that may be easier to apply. (Ans: κ = |dT /dt ds/dt |, where ds/dt = |r′(t)|).
  3. Consider the curve r(t) = 〈 t 33 , t^2 , 2 t〉. (a) Find the unit tangent vector at the point P( 83 , 4 , 4). (Ans: 〈 23 , 23 , 13 〉) (b) Use the formula you derived in Problem 2 to find the curvature of the curve at P. (Ans: 181 )
  4. Find the value of a for which the parabola y = ax^2 has curvature 4 at the origin, as follows. (a) Parametrize the parabola by a parameter t. (b) Use the formula you derived in Problem 2 to find the curvature at the origin in terms of a. (Ans: 2 a) (c) Answer the question. (Ans: a = 2) (d) Check your answer: using your calculator, draw a graph of the resulting parabola (y = 2x^2 ) and the osculating circle of curvature 4 at the origin.

Functions of several variables

  1. Chapter 15 Review, Concept Check: 1, Exercises: 1-8.

Other

  1. Chapter 13 Review, Concept Check: 1,3,5,7,8,
  2. Chapter 13 Review, True-False 1-