MATH 271 Summer 2016
Midterm
August 5, 2016, at 10:00am
Time limit: 110 minutes
Instructor: Mark Girard
Name:
ID number:
TA:
This exam contains 7 pages (including this cover page) and 6 problems. Check to see if any pages
are missing. Enter all requested
MATH 271 Summer 2016
Study Guide for Final Exam
Topics list
Proof techniques
How to prove different types of statements (i.e. , , if. . . then, or, and)
Negations, converses, contrapositives
Direct proof, proof by contradiction, proof by counterexample
P
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UNIVERSITY OF CALGARY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS AND STATISTICS
MIDTERM EXAMINATION
MATH 249 - WINTER 2015
DATE: FEBRUARY 11, 2015 at 19:00
I.D. NUMBER
LAST NAME
TIME: 2 HOURS
FIRST NAME
LAB #
EXAMINATION RULES
1. This is a
w CALGARY
UNIVERSITY OF
UNIVERSITY OF CALGARY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS 85 STATISTICS
FINAL EXAM (PRACTICE B)
' MATH 265
EXAMINATION RULES
1.
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This is a closed book examination.
Calculators are not permitte
UNIVERSITY OF
V CALGARY
UNIVERSITY OF CALGARY
FACULTY OF SCIENCE _
DEPARTMENT OF MATHEMATICS &. STATISTICS
MIDTERM 1 (PRACTICE A)
MATH 265
EXAMINATION RULES
1. This is a closed book exmninntion.
2. Calculators are not permitted.
. The uee of personal el
UNIVERSITY OF
CALGARY
UNIVERSITY OF CALGARY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS & STATISTICS
FINAL EXAM (PRACTICE A)
MATH 265
mm
EXAMINATION RULES
1. This is a closed book examination.
2. Calculators are not permitted.
The use of personal electro
UNIVERSITY OF CALGARY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS & STATISTICS
MIDTERM 2 (PRACTICE A)
MATH 265
NAME
STUDENT ID
EXAMINATION RULES
1. This is a closed book examination.
2. Calculators are not permitted.
3. The use of personal electronic or
UNIVERSITY OF CALGARY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS & STATISTICS
MIDTERM 1 (PRACTICE B)
MATH 265
NAME
STUDENT ID
EXAMINATION RULES
1. This is a closed book examination.
2. Calculators are not permitted.
3. The use of personal electronic or
ELECTROMAGNETISMO: DE LA CIENCIA A LA TECNOLOGA
Autor: ELIEZER BRAUN
COMIT DE SELECCIN
EDICIONES
I. INTRODUCCIN
II. LA ELECTRICIDAD HASTA EL AO 1800
III. EL MAGNETISMO HASTA EL AO 1800
IV. HAY RELACIN ENTRE ELECTRICIDAD Y MAGNETISMO?
V. FARADAY. LA INDUCC
An Example of Motivating
Mathematical Induction for Computer
Science
The Chinese Rings or Patience Puzzle
A Recursive Method to Remove
Rings and Unlock the Puzzle
To Remove the n rings:
Reduce the puzzle to an n-1 ring puzzle.
Remove the leftmost n-1 ring
NP is the set of all problems solvable in polynomial time by a non-deterministic program. Yikes, what
does that mean? Wait until the algorithms course. But basically, it means that you can verify a guess of
the solution in polynomial time. Non-determinism
Math 249 - Week 8 (Oct. 31 to Nov. 4) Lab Exercises - Fall 2016
1. For the following curves, use implicit differentiation to find
line for the curve at the given point.
dy
dx .
Determine the tangent
(a)
x2 y 2 = 3x 2y 3 at (1, 1)
ANSWER: Differentiating b
Math 249 - Week 7 (Oct. 24 to Oct. 28) Lab Exercises - Fall 2016
1
1. (a) Suppose f (x) = x+1
. Using the limit definition of the derivative, calculate f 0 (1).
Find the tangent line for f at x = 1.
ANSWER: Of course, now we have rules to compute derivati
Math 249 - Week 9 and 10 (Nov. 7 to Nov. 9; Nov. 14 to Nov. 18) Lab
Exercises - Fall 2016
1. Alice and Bob are standing in the same location. Alice begins walking north at 5km/h
and Bob begins walking west at 4km/h. How quickly are they moving apart at ti
Math 249 - Week 8 (Oct. 31 to Nov. 4) Lab Exercises - Fall 2016
1. For the following curves, use implicit differentiation to find
line for the curve at the given point.
dy
dx .
Determine the tangent
(a)
x2 y 2 = 3x 2y 3 at (1, 1)
(b)
y 4 = ex
2 y 2
at (1,
Math 249 - Week 11 (Nov.21 to Nov. 25) Lab Exercises - Fall 2016
1. Use LHopitals rule on the following limits:
(a)
2x + e4x
.
x 3 ex
lim
ANSWER: This is of the form /00 . Apply LH to get
2x + e4x
2 + 4e4x
=
lim
.
x 3 ex
x
ex
lim
We still have the same in
Math 249 - Week 11 (Nov.21 to Nov. 25) Lab Exercises - Fall 2016
1. Use LHopitals rule on the following limits:
(a)
2x + e4x
.
x 3 ex
lim
(b)
lim x ln(1 + 3/x).
x
(c)
lim (x 1) tan( x).
2
x1+
2. For the following functions, find all intervals of increase/
Math 249 - Week 7 (Oct. 24 to Oct. 28) Lab Exercises - Fall 2016
1
1. (a) Suppose f (x) = x+1
. Using the limit definition of the derivative, calculate f 0 (1).
Find the tangent line for f at x = 1.
(b) Suppose f (x) = 2x 1. Using the limit definition of
Math 249 - Week 9 and 10 (Nov. 7 to Nov. 9; Nov. 14 to Nov. 18) Lab
Exercises - Fall 2016
1. Alice and Bob are standing in the same location. Alice begins walking north at 5km/h
and Bob begins walking west at 4km/h. How quickly are they moving apart at ti
Math 249 - Week 2 (Sept. 19 to Sept. 23) Lab Exercises - Fall 2016
1. Simplify the following expressions into the form xa y b .
(a)
x3 y 2
.
x2 y 4
ANSWER: x5 y 2 .
(b)
p
x4 y 5
.
4
y 2 x3
ANSWER: x11/4 y 9/2 .
2. Write the following rational expressions
Math 249 - Week 3 (Sept. 26 to Sept. 30) Lab Exercises - Fall 2016
1. By rewriting the following equations in the correct form, identity each as a circle or
parabola. Graph the result.
(a) x2 4x + 10y = 0.
ANSWER: Completing the square gives y
parabola o
Math 249 - Week 4 (Oct. 3 to 7) Lab Exercises - Fall 2016
1. Find the inverses of the following functions. Try to graph the functions (by translating
the base function) and then the inverse by reflection over the line y = x.
(a) y = 2(x 1)3 3.
ANSWER: Swi
Math 249 - Week 5 (Oct. 11 to Oct. 15) Lab Exercises - Fall 2016
1. By removing any common factors, compute the following limits.
(a) limx1
x2 +x2
.
3x2 +2x5
(b) limx2
x3 +x2 6x
.
x2
(c) limx2
x4 16
.
x 2
2. By conjugating, compute the following limits.
Math 249 - Week 4 (Oct. 3 to 7) Lab Exercises - Fall 2016
1. Find the inverses of the following functions. Try to graph the functions (by translating
the base function) and then the inverse by reflection over the line y = x.
(a) y = 2(x 1)3 3.
(b) y = tan