Midpoint - Calculus - Exam, Exams of Calculus

This is the Exam of Calculus Indefinite Integrals, Limits, Explanation, Curve Parametrized, Involve the Variables etc. Key important points are: Midpoint Rule, Indefinite Integrals, Methods, Approximate, Large, Choose, Accurate, Boxes, Trigonometric Substitution, Error Bound

Typology: Exams

2012/2013

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Simon Fraser University
MATH 251 - Summer 2005
Midterm 1
June 1, 2005, 8:30 – 9:20 am
Last Name (please print): _________________________________________
First Name (please print): _________________________________________
Student Number: _________________________________________
Signature: _________________________________________
Instructions:
1. DO NOT OPEN THIS BOOKLET
UNTIL TOLD TO DO SO.
2. Fill in the above box.
3. This exam contains 6 pages with a total
of 6 questions. Once the exam begins
please check to make sure your exam
booklet is complete.
4. Only complete well-organized solution
will receive full credit
5. If you run out of space in a problem, use
the space on the back of the previous
page and clearly indicate where the
solution continues.
6. Only scientific calculators are allowed.
7. No book, paper, or device, other than the
usual writing instruments, this booklet
and a scientific calculator, shall be
within reach of a student during the
examination.
8. During the examination, speaking to,
communicating with, or deliberately
exposing written papers to the view of
other examinees is forbidden.
Question Marks
1 /8
2 /6
3 /4
4 /4
5 /10
6 /8
Total /40
1
pf3
pf4
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Simon Fraser University

MATH 251 - Summer 2005

Midterm 1 June 1, 2005, 8:30 – 9:20 am

Last Name (please print): _________________________________________

First Name (please print): _________________________________________

Student Number: _________________________________________

Signature: _________________________________________

Instructions:

  1. DO NOT OPEN THIS BOOKLET UNTIL TOLD TO DO SO.
  2. Fill in the above box.
  3. This exam contains 6 pages with a total of 6 questions. Once the exam begins please check to make sure your exam booklet is complete.
  4. Only complete well-organized solution will receive full credit
  5. If you run out of space in a problem, use the space on the back of the previous page and clearly indicate where the solution continues.
  6. Only scientific calculators are allowed.
  7. No book, paper, or device, other than the usual writing instruments, this booklet and a scientific calculator, shall be within reach of a student during the examination.
  8. During the examination, speaking to, communicating with, or deliberately exposing written papers to the view of other examinees is forbidden.

Question Marks

Total /

a) Suppose that M is the midpoint of the segment AB in space and that P is another point. Show that 1 ( 2

PM = PA + PB )

JJJJG JJJG JJJG

. [4 marks]

b) Suppose that a^ K^ and b are two unit vectors. Show that

K

a + b

K K

is perpendicular to ab. [4 marks]

K K

  1. Determine whether or not the four points A (2,0, −3) , B (0,5,4), C (1,1, −1) and D (5, −12, −18)are coplanar. [4 marks]
  2. A particle moves along a curve C with velocity given by v t^ K( )^ =< 2 , t e t ,sin( π t )>. At time t = 1 , the particle passes through the point P (1,0,1/ π ). Determine the position vector r t^ K( ) at time t. [4 marks]
  1. A space curve is given by the vector function r t^ K( )^ =< 4 , t t^2 ,2 t^2 >.

a) Find the equation of the line that is tangent to its position vector and that is parallel to the line: 1 2 2

x − (^) = y = z + (^1). [6 marks]

b) Find the curvature of curve κ for t ≥ 0.

( Hint: 1 |^ '( )^ ''( ) | 3 | '( ) |

dT dT r t r t ds v dt r t

κ = = = ×

K K K K

K ).^ [4 marks]