Difference between revisions of "BSc: Mathematical Analysis I.f22"

From IU
Jump to navigation Jump to search
 
(12 intermediate revisions by one other user not shown)
Line 11: Line 11:
 
|-
  
|-
 
! Section !! Topics within the section
  
! Section !! Topics within the section
  +
|-
  +
| Limits ||
  +
# Limits of Sequences
  +
# Newton's Method
  +
# Limits of Functions
 
|-
  
|-
 
| Derivatives ||
 
| Derivatives ||
Line 21: Line 26:
 
# Inverse Functions and Differentiation
  
# Inverse Functions and Differentiation
 
# Differentiation of the Trigonometric, Exponential and Logarithmic Functions
  
# Differentiation of the Trigonometric, Exponential and Logarithmic Functions
  +
# L'Hopital's Rule
 
# Increasing and Decreasing Functions
  
# Increasing and Decreasing Functions
 
# The Second Derivative and Concavity
  
# The Second Derivative and Concavity
Line 37: Line 43:
 
# Parametric Curves
  
# Parametric Curves
 
# Applications of the integrals
  
# Applications of the integrals
−
|-
  
−
| Limits ||
 
−
# Limits of Sequences
  
−
# Newton's Method
  
−
# Limits of Functions
  
−
# L'Hopital's Rule
  
 
# Improper integrals
  
# Improper integrals
 
|}
  
|}
  +
 
== Intended Learning Outcomes (ILOs) ==
  
== Intended Learning Outcomes (ILOs) ==
   
Line 87: Line 88:
 
! Grade !! Range !! Description of performance
  
! Grade !! Range !! Description of performance
 
|-
  
|-
−
| A. Excellent || 90-100 || -
 +
| A. Excellent || 85-100 || -
 
|-
  
|-
−
| B. Good || 75-89 || -
 +
| B. Good || 70-84 || -
 
|-
  
|-
−
| C. Satisfactory || 60-74 || -
 +
| C. Satisfactory || 50-69 || -
 
|-
  
|-
−
| D. Fail || 0-59 || -
 +
| D. Fail || 0-49 || -
 
|}
  
|}
   
Line 104: Line 105:
 
| Midterm || 20
  
| Midterm || 20
 
|-
  
|-
−
| Quizzes || 28 (2 for each)
 +
| Tests || 28 (14 for each)
 
|-
  
|-
 
| Final exam || 50
  
| Final exam || 50
Line 119: Line 120:
   
 
=== Open access resources ===
  
=== Open access resources ===
−
* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985
 +
* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985 [https://www.cds.caltech.edu/~marsden/volume/Calculus/ link]
 
* Zorich, V. A. Mathematical Analysis I, Translator: Cooke R. (2004)
  
* Zorich, V. A. Mathematical Analysis I, Translator: Cooke R. (2004)
   
  +
= Teaching Methodology: Methods, techniques, & activities =
−
=== Software and tools used within the course ===
  
−
* No.
  
   
 
== Activities and Teaching Methods ==
  
== Activities and Teaching Methods ==
Line 131: Line 131:
 
! Teaching Techniques !! Section 1 !! Section 2 !! Section 3
  
! Teaching Techniques !! Section 1 !! Section 2 !! Section 3
 
|-
  
|-
−
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) || 0 || 0 || 0
 +
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) || 1 || 1 || 1
 
|-
  
|-
−
| Project-based learning (students work on a project) || 1 || 1 || 1
 +
| Project-based learning (students work on a project) || 0 || 0 || 0
 
|-
  
|-
 
| Modular learning (facilitated self-study) || 0 || 0 || 0
  
| Modular learning (facilitated self-study) || 0 || 0 || 0
Line 174: Line 174:
 
| Development of individual parts of software product code || 0 || 0 || 0
  
| Development of individual parts of software product code || 0 || 0 || 0
 
|-
  
|-
−
| Individual Projects || 1 || 1 || 1
 +
| Individual Projects || 0 || 0 || 0
 
|-
  
|-
 
| Group projects || 0 || 0 || 0
  
| Group projects || 0 || 0 || 0
Line 180: Line 180:
 
| Flipped classroom || 0 || 0 || 0
  
| Flipped classroom || 0 || 0 || 0
 
|-
  
|-
−
| Quizzes (written or computer based) || 0 || 0 || 0
 +
| Quizzes (written or computer based) || 1 || 1 || 1
 
|-
  
|-
 
| Peer Review || 0 || 0 || 0
  
| Peer Review || 0 || 0 || 0
Line 186: Line 186:
 
| Discussions || 1 || 1 || 1
  
| Discussions || 1 || 1 || 1
 
|-
  
|-
−
| Presentations by students || 1 || 1 || 1
 +
| Presentations by students || 0 || 0 || 0
 
|-
  
|-
 
| Written reports || 0 || 0 || 0
  
| Written reports || 0 || 0 || 0
Line 202: Line 202:
   
 
==== Section 1 ====
  
==== Section 1 ====
  +
# Find limits of the following sequences or prove that they do not exist:
−
# A steel company must decide how to allocate next week’s time on a rolling mill, which is a machine that takes unfinished slabs of steel as input and can produce either of two semi-finished products: bands and coils. The mill’s two products come off the rolling line at different rates:
  
  +
#* <math>a_n=n-\sqrt{n^2-70n+1400}</math>;
−
#:: Bands 200 tons/hr
  
  +
#* <math display="inline">d_n=\left(\frac{2n-4}{2n+1}\right)^{n}</math>;
−
#:: Coils 140 tons/hr.
  
  +
#* <math display="inline">x_n=\frac{\left(2n^2+1\right)^6(n-1)^2}{\left(n^7+1000n^6-3\right)^2}</math>.
−
#: They also produce different profits:
  
−
#:: Bands $ 25/ton
  
−
#:: Coils $ 30/ton.
  
−
#: Based on currently booked orders, the following upper bounds are placed on the amount of each product to produce:
  
−
#:: Bands 6000 tons
  
−
#:: Coils 4000 tons.
  
−
#: Given that there are 40 hours of production time available this week, the problem is to decide how many tons of bands and how many tons of coils should be produced to yield the greatest profit. Formulate this problem as a linear programming problem. Can you solve this problem by inspection?
  
−
# Solve the following linear programming problems.
  
−
#: maximize <math>6x_1+ 8x_2 + 5x_3 + 9x_4 </math>
  
−
#: subject to <math>2x_1+ x_2 + x_3 + 3x_4 \leq 5 </math>
  
−
#:: <math>x_1 + 3x_2 + x_3 + 2x_4 \leq 3 </math>
  
−
#:: <math>x1, x2, x3, x4 \geq 0 </math>
  
−
# Give an example showing that the variable that becomes basic in one iteration of the simplex method can become nonbasic in the next iteration.
  
−
# Solve the given linear program using the dual–primal two phase algorithm.
  
−
#: maximize <math>2x_1 - 6x_2</math>
  
−
#: subject to <math>-x_1 - x_2 - x_3 \leq -2</math>
  
−
#:: <math>2x_1 - x_2 + x_3 \leq 1 </math>
  
−
#:: <math>x1, x2, x3 \geq 0 </math>
  
   
 
==== Section 2 ====
  
==== Section 2 ====
  +
# A plane curve is given by <math>x(t)=-\frac{t^2+4t+8}{t+2}</math>, <math display="inline">y(t)=\frac{t^2+9t+22}{t+6}</math>. Find
−
# Find the minimum of the function <math> f = \lambda / log \lambda </math> using the following methods:
  
  +
#: the asymptotes of this curve;
−
#* Newton method
  
  +
#: the derivative <math display="inline">y'_x</math>.
−
#* Quasi-Newton method
  
  +
# Apply Leibniz formula Find <math display="inline">y^{(n)}(x)</math> if <math display="inline">y(x)=\left(x^2-2\right)\cos2x\sin3x</math>.
−
#* Quadratic interpolation method
  
  +
#: Draw graphs of functions
−
# State possible convergence criteria that can be used in direct search methods.
  
  +
#: Find asymptotes
−
# Why is the steepest descent method not efficient in practice, although the directions used are the best directions?
  
  +
# Find the derivatives of the following functions:
−
# What is the difference between quadratic and cubic interpolation methods?
  
  +
#* <math display="inline">f(x)=\log_{|\sin x|}\sqrt[6]{x^2+6}</math>;
−
# Why is refitting necessary in interpolation methods?
  
  +
#* <math display="inline">y(x)</math> that is given implicitly by <math display="inline">x^3+5xy+y^3=0</math>.
−
# What is a direct root method?
  
−
# What is the basis of the interval halving method?
  
−
# What is the difference between Newton and quasi-Newton methods?
  
   
 
==== Section 3 ====
  
==== Section 3 ====
−
# Solve the following LP problem by dynamic programming:
 +
# Find the following integrals:
  +
#*<math display="inline">\int\frac{\sqrt{4+x^2}+2\sqrt{4-x^2}}{\sqrt{16-x^4}}\,dx</math>;
−
#: Maximize <math> f (x_1, x_2) = 10x_1 + 8x_2</math>
  
−
#: subject to <math> 2x_1 + x_2 \leq 25</math>
 +
#*<math display="inline">\int2^{2x}e^x\,dx</math>;
−
#:: <math> 3x_1 + 2x_2 \leq 45</math>
 +
#*<math display="inline">\int\frac{dx}{3x^2-x^4}</math>.
  +
# Find the indefinite integral <math display="inline">\displaystyle\int x\ln\left(x+\sqrt{x^2-1}\right)\,dx</math>.
−
#:: <math> x_2 \leq 10 </math>
  
  +
# Find the length of a curve given by <math display="inline">y=\ln\sin x</math>, <math display="inline">\frac{\pi}4\leqslant x\leqslant\frac{\pi}2</math>.
−
#:: <math> x1, x2 \geq 0</math>
  
−
#: Verify your solution by solving it graphically.
  
−
# Consider the following tree solution for a minimum cost network flow problem:
  
−
#:[[Image:ItO_p1.png]]
  
−
#:As usual, bold arcs represent arcs on the spanning tree, numbers next to the bold arcs are primal flows, numbers next to non-bold arcs are dual slacks, and numbers next to nodes are dual variables.
  
−
#* For what values of is this tree solution optimal?
  
−
#* What are the entering and leaving arcs?
  
−
#* After one pivot, what is the new tree solution?
  
−
#* For what values of is the new tree solution optimal?
  
   
 
=== Final assessment ===
  
=== Final assessment ===
−
'''Section 1'''
  
−
# Solve the following linear programming problems.
  
−
#: maximize <math>x_1 + 3x_2</math>
  
−
#: subject to <math>-x_1 - x_2 \leq -3</math>
  
−
#:: <math>-x_1 + x_2 \leq -1</math>
  
−
#:: <math>-x_1 + 2x_2 \leq 2</math>
 
−
#:: <math>x1, x2 \geq 0</math>
 
−
# Give an example showing that the variable that becomes basic in one iteration of the simplex method can become nonbasic in the next iteration.
  
−
# Solve the given linear program using the dual–primal two phase algorithm.
  
−
#: maximize <math>6x_1 + 8x_2 + 5x_3 + 9x_4</math>
  
−
#: subject to <math>x_1 + x_2 + x_3 + x_4 = 1</math>
  
−
#:: <math>x_1, x_2, x_3, x_4 \geq 0</math>
 
   
−
'''Section 2'''
 +
==== Section 1 ====
  +
# Find a limit of a sequence
−
# Perform two iterations of the steepest descent method to minimize the function given from the stated starting point <math>(-1.2, 1)</math>
  
  +
# Find a limit of a function
−
#:: <math>f(x_1, x_2) = 100(x_2 - x_1^2 )^2 + (1 - x_1)^2</math>
  
−
# Minimize <math>f = 2x_1^2 + x_2^2</math> by using the steepest descent method with the starting point <math>(1, 2)</math> (two iterations only).
  
−
# Minimize <math>f = x_1^2 + 3x_2^2 + 6x_3^2</math> by Newton's method using the starting point as <math>(2,-1, 1)</math> .
  
   
−
'''Section 3'''
 +
==== Section 2 ====
  +
# Apply the appropriate differentiation technique to a given problem.
−
* It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below. Find the investment policy for maximizing the total electric power generated.
  
  +
# Find a derivative of a function
−
{| class="wikitable" style="height:14em;"
  
  +
# Apply Leibniz formula
−
|-
  
  +
# Draw graphs of functions
−
! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3
  
  +
# Find asymptotes of a parametric function
−
|-
  
  +
−
| R0(x) || 0 || 0 || 0
  
  +
==== Section 3 ====
−
|-
 
  +
# Apply the appropriate integration technique to the given problem
−
| R1(x) || 2 || 1 || 3
  
  +
# Find the value of the devinite integral
−
|-
  
  +
# Calculate the area of the domain or the length of the curve
−
| R2(x) || 4 || 5 || 5
  
−
|-
  
−
| R3(x) || 6 || 6 || 6
  
−
|}
  
−
* A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.
  
−
* Consider the following tree solution for a minimum cost network flow problem:
  
−
*: [[Image:ItO_p2.png]]
  
−
*: As usual, bold arcs represent arcs on the spanning tree, numbers next to the bold arcs are primal flows, numbers next to non-bold arcs are dual slacks, and numbers next to nodes are dual variables.
  
−
*:: For what values of is this tree solution optimal?
  
−
*:: What are the entering and leaving arcs?
  
−
*:: After one pivot, what is the new tree solution?
  
−
*:: For what values of is the new tree solution optimal?
  
   
 
=== The retake exam ===
  
=== The retake exam ===

Latest revision as of 11:00, 11 November 2022

Mathematical Analysis I

  • Course name: Mathematical Analysis I
  • Code discipline: CSE201
  • Subject area: Math

Short Description

This calculus course covers differentiation and integration of functions of one variable, with applications. The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Should be understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation.

Course Topics

Course Sections and Topics
Section Topics within the section
Limits
  1. Limits of Sequences
  2. Newton's Method
  3. Limits of Functions
Derivatives
  1. Derivative as a Limit
  2. Leibniz Notation
  3. Rates of Change
  4. The Chain Rule
  5. Fractional Powers and Implicit Differentiation
  6. Related Rates and Parametric Curves
  7. Inverse Functions and Differentiation
  8. Differentiation of the Trigonometric, Exponential and Logarithmic Functions
  9. L'Hopital's Rule
  10. Increasing and Decreasing Functions
  11. The Second Derivative and Concavity
  12. Maximum-Minimum Problems
  13. Graphing
Integrals
  1. Sums and Areas
  2. The Fundamental Theorem of Calculus
  3. Definite and Indefinite Integrals
  4. Integration by Substitution
  5. Changing Variables in the Definite Integral
  6. Integration by Parts
  7. Trigonometric Integrals
  8. Partial Fractions
  9. Parametric Curves
  10. Applications of the integrals
  11. Improper integrals

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

This calculus course will provide an opportunity for participants to:

  • understand key principles involved in differentiation and integration of functions
  • solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities
  • become familiar with the fundamental theorems of Calculus
  • get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation.

ILOs defined at three levels

We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

  • remember the differentiation techniques
  • remember the integration techniques
  • remember how to work with sequences and series

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

  • apply the derivatives to analyse the functions
  • integrate
  • understand the basics of approximation

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

  • Take derivatives of various type functions and of various orders
  • Integrate
  • Apply definite integral
  • Expand functions into Taylor series
  • Apply convergence tests

Grading

Course grading range

Grade Range Description of performance
A. Excellent 85-100 -
B. Good 70-84 -
C. Satisfactory 50-69 -
D. Fail 0-49 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Midterm 20
Tests 28 (14 for each)
Final exam 50
In-class participation 7 (including 5 extras)

Recommendations for students on how to succeed in the course

  • Participation is important. Attending lectures is the key to success in this course.
  • Review lecture materials before classes to do well.
  • Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.

Resources, literature and reference materials

Open access resources

  • Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985 link
  • Zorich, V. A. Mathematical Analysis I, Translator: Cooke R. (2004)

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Teaching and Learning Methods within each section
Teaching Techniques Section 1 Section 2 Section 3
Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) 1 1 1
Project-based learning (students work on a project) 0 0 0
Modular learning (facilitated self-study) 0 0 0
Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) 1 1 1
Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) 0 0 0
Business game (learn by playing a game that incorporates the principles of the material covered within the course) 0 0 0
Inquiry-based learning 0 0 0
Just-in-time teaching 0 0 0
Process oriented guided inquiry learning (POGIL) 0 0 0
Studio-based learning 0 0 0
Universal design for learning 0 0 0
Task-based learning 0 0 0
Activities within each section
Learning Activities Section 1 Section 2 Section 3
Lectures 1 1 1
Interactive Lectures 1 1 1
Lab exercises 1 1 1
Experiments 0 0 0
Modeling 0 0 0
Cases studies 0 0 0
Development of individual parts of software product code 0 0 0
Individual Projects 0 0 0
Group projects 0 0 0
Flipped classroom 0 0 0
Quizzes (written or computer based) 1 1 1
Peer Review 0 0 0
Discussions 1 1 1
Presentations by students 0 0 0
Written reports 0 0 0
Simulations and role-plays 0 0 0
Essays 0 0 0
Oral Reports 0 0 0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

  1. Find limits of the following sequences or prove that they do not exist:
    • ;
    • ;
    • .

Section 2

  1. A plane curve is given by , . Find
    the asymptotes of this curve;
    the derivative .
  2. Apply Leibniz formula Find if .
    Draw graphs of functions
    Find asymptotes
  3. Find the derivatives of the following functions:
    • ;
    • that is given implicitly by .

Section 3

  1. Find the following integrals:
    • ;
    • ;
    • .
  2. Find the indefinite integral .
  3. Find the length of a curve given by , .

Final assessment

Section 1

  1. Find a limit of a sequence
  2. Find a limit of a function

Section 2

  1. Apply the appropriate differentiation technique to a given problem.
  2. Find a derivative of a function
  3. Apply Leibniz formula
  4. Draw graphs of functions
  5. Find asymptotes of a parametric function

Section 3

  1. Apply the appropriate integration technique to the given problem
  2. Find the value of the devinite integral
  3. Calculate the area of the domain or the length of the curve

The retake exam

Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.

Retrieved from "https://eduwiki.innopolis.university/index.php?title=BSc:_Mathematical_Analysis_I.f22&oldid=7765"