Course presentation
- Name, syllabus;
- Plan:
1) Present resources; objectives (direct/indirect benefits)
2) How the course is organized
3) Using technology
4) Background information (not test);
…
1) - Objectives:
1) To introduce the basic ideas of real analysis with proofs;
- Will involve some algebraic structures;
- The 1st part (Ch.2-3) is also a general introduction to Topology (metric);
2) To teach students how to designing mathematical proofs, starting with:
- Idea: picture (graphical representation);
- Plan: “flow chart” of the proof’s logic;
- Math code: the details of the proof (the program in “low level” math language).
This would help students think conceptually, intuitively and globally, but read and write rigorous
mathematics.
3) To help students develop a rigorous understanding of the theorems of calculus;
4) To enhance student’s ability to communicate abstract ideas (written & oral form);
- Course format:
- Course format is “Lecture – Recitation”: more details …
- Learning cycle:
- Basic idea: repeat a topic to learn it;
- Read before lecture; write an outline; work HW problems; read next section before lect.
- “Journal
of activities”: testing it to be used w. Calc. students;
- Please provide feedback on its utility
- Class activities:
- Answering questions;
- Student practice: proofs; no grading; 1 pt. for turning them in (reasonably completed);
- Presenting solutions (depending on time / # students);
2) Background information;
3) 5 min break;
4) Sections 1.1 & 1.2
Resources for Calc I
 Instructor: Lucian Ionescu
- Office hours: MWRF 1:00-2:00 p.m., STV 303G;
- Appointments: in person or by e-mail;
- Web site: LI homepage/mat145
 Text: Thomas’ Calculus (Early transcendentals), 11th ed.
Content:
Ch.1 – Functions as mathematical models
(Calculus 1,2,3 … “theme”)
Ch.2,3,4,[5] – Studying functions (models):
- Limits, derivatives, integration and applications;
Objectives:
- General: building knowledge and learning skills;
- Specific: calculus methods applied to sciences.
 Calculator with a Computer Algebra System (CAS): TI-89/92
(Symbolic differentiation, integration, solving D.E. etc.)
- Solving a problem: understanding versus computing.
 Syllabus
1) Lecture – recitation format
2) Learning cycle
3) List of homework assignments; timing …
4) Evaluation / grading scale
 Homework:
- Solutions to problems corresponding to sections covered M-F;
- Before exams, complete homework by Monday (Example)
The Learning Cycle
 What is “learning”?
- Acquisition of Knowledge
- Building Skills.
 How “to learn”? (Many “good answers” possible!)
1. BEFORE class, skim through the assigned section (1/2 hour)
(Know “what’s on the menu” - next class).
2. Come to class and:
- Answer questions from the previous section;
- Ask questions from the current section;
- Take class notes; do not just “follow the explanations”;
- Focus on concepts and methods.
3. After class,
- Review the material, write an OUTLINE of the section;
- Attempt all the homework problems:
a) First, without the help of the book;
b) If it doesn’t work, use the book (Review as above);
c) Write a list of questions to ask (group/instructor).
 Impact on grade (usually): 1-4 leads to A!
“Calculus: 1,2,3…”
Real System
- Quantities
Mathematical Model
Fit a ...
- Variables
- Relations
- Equations
- Questions
Solve
Answers(s)
Interpret
Result(s)
Solving a Problem:
- Find the quantities involved-> variables;
- Find the relations among variables-> equations;
- Translate the questions into tasks (solving eq.);
- Apply the appropriate method;
- Interpret the result.
Ch. 1 Functions
1.1 functions and Their Graphs
- Modeling correlations: static (sides and area), dynamic:
deterministic => idea of function (x->y)
- Keywords: function, domain, codomain, rule, range,
graph
- Operations: sketching a graph, evaluating a function
from its graph, representing a function numerically
- Piecewise-defined functions
- Summary:
- Function: domain, codomain, rule/graph
- Presentations: graphical, numerical, by formula
1.2 Identifying Functions; Mathematical Models
- Classes of functions: linear, power functions,
polynomials, rational functions, transcendental
(exponential, logarithmic, trigonometric)
- Increasing, decreasing functions
- Even/odd functions, symmetry
- Mathematical models
- Proportionality; Kepler’s 3rd Law
1.3 Combining Functions; Shifting and Scaling Graphs
- Combining functions: +,-,x,:
- Composition of functions
- “Decomposing” a function (viewing a function as a
composition)
- Shifting a Graph of a function
- Scaling (stretching and compressing)
- Reflecting
- Ellipses; standard equation of an ellipse
- Transformations of trigonometric graphs
1.4 Graphing with calculators and Computers
- Windows
- Regression analysis
1.5 Exponential Functions
- Exponential behavior
- exp
- Exponential growth and decay
- Examples: savings accounts, radioactive decay
1.6 Inverse functions and logarithms
- 1:1, horizontal line test;
- Inverse functions; finding inverses
- Logarithmic functions; properties
- Applications: half-life
- Inverse trigonometric functions
Ch. 2 Limits and Continuity
2.1 Rates of Change and Limits
- Average and instantaneous speed
- Average rates of change and secant lines
- Limits of function values
- Using calculators to find limits
2.2 Calculating limits using the limit laws
- Limit laws
- Examples
- The sandwich theorem
2.3 The Precise Definition of Limit
- Example 1: a Linear Function
- Definition of limit
- Examples: testing the definition
- Finding deltas algebraically for given epsilons
- Using the definition to prove theorems
- Limit law for addition
2.4 One-Sided Limits and Limits at Infinity
- Extending the concept of limits to one-sided Limits
- LH-limit, RH-limit
- Theorem: limit  LH & RH
- Precise definition of one-sided limits
- Limits involving sin t / t
- Finite limits as x-> infinity
- Limit Laws
- Limits at infinity of rational functions
- Horizontal asymptotes
- The Sandwich Theorem Revisited
- Oblique asymptotes
2.5 Infinite Limits and Vertical Asymptotes
- Infinite Limits
- Precise definition of infinite limits
- Vertical asymptotes
- Dominant terms
2.6 Continuity
- Continuity at a point
- Continuity test
- Continuous functions: definition, properties (+,-,x,:)
- Composition of continuous functions
- Continuous extension to a point
- Intermediate Value Theorem for Continuous Functions
2.7 Tangents and Derivatives
- What is a tangent to a curve?
- Finding a tangent to a graph of a function
- Slope, tangent line
- Rates of change: derivative at a point.
- Summary
Ch. 3 Differentiation
3.1 The Derivative as a Function
- Calculating Derivatives from the Definition
- Graphing the Derivative
- Differentiable on an interval; one-sided derivatives
- When a function does NOT have a derivative
- Differentiable functions are continuous
- The intermediate value property for derivatives
3.2 Differentiation rules for Polynomials, Exponentials,
Products and Quotients
- Powers, multiples, +, - Derivative of exponential functions
- Products and quotients
- 2nd and higher-order derivatives
3.3 The Derivative as a Rate of Change
- Instantaneous rate of change
- Motion along a line: position, speed, acceleration
- Derivatives in economics
- Sensitivity to change
3.4 Derivatives of trigonometric functions
- derivative of sine function, cosine
- Simple harmonic motion
- Derivatives of other trig functions
3.5 The Chain Rule and Parametric Equations
- Derivative of a composite function
- The Chain Rule; “outside-Inside rule”
- Repeated use of the Chain Rule
- The Chain Rule with powers of a Function
- Parametric equations
- Slopes of parametrized curves
3.6 Implicit Differentiation
- Implicitly Defined Functions
- Implicite Differentiation
- Lenses, Tangents, and Normal Lines
- Derivatives of higher order
- Rational powers of differentiable functions
3.7 Derivatives of Inverse Functions and Logarithms
- Derivatives of Inverses of Differentiable Functions
- Parametrizing inverse functions
- Derivatives of ln(x), a^u(x), log_a(u(x))
- Logarithmic differentiation
- The Power Rule (general form)
- Number e expressed as a limit
3.8 Inverse Trigonometric Functions
- Inverses of: tan, cot, sec, csc;
- Derivatives of inverses: sin^-1(x) etc.
3.9 Related Rates
- Related rates equations
- Related rates problem strategy
3.10 Linearization and Differentials
- Linearization
- Standard Linear approximation
- Differentials
- Estimating with Differentials
- Error in differential approximations
- Sensitivity to change
Ch. 4 Applications of Derivatives
4.1 Extreme Values of Functions
- Absolute maximum, absolute minimum
- Local (relative) extreme values
- Finding extrema
- The 1st derivative theorem for local extreme val.
- Critical point
- How to find the absolute extrema of a continuous
function on a finite closed interval
4.2 The Mean Value Theorem
- Rolle’s Theorem
- The Mean Value Theorem
- A Physical Interpretation
- Mathematical Consequences
- Proofs of the Laws of Logarithms
- Laws of Exponents
- Finding velocity and position from acceleration
4.3 Monotonic Functions and the First Derivative Test
- Increasing Functions and Decreasing Functions
- 1st Derivative Test for Monotonic Functions
- 1st Derivative Test for Local Extrema
4.4 Concavity and Curve Sketching
- Concavity: definition, 2nd derivative test
- Points of inflection
- 2nd Derivative Test for Local Extrema
- Strategy for Graphing y=f(x)
- Learning about functions from derivatives
4.5 Applied Optimization Problems
- Examples from Business and Industry
- Solving Applied Optimization Problems
- Examples from Mathematics and Pfysics
- Examples from Economics
4.6 Indeterminate Forms and L’Hospital’s Rule
- Indeterminate form 0/0
- L’Hopital’s Rule
- Cauchy Mean Value Theorem
- Using L’Hopital’s Rule
- Indeterminate forms: inf/inf, inf x 0, inf-inf
- Indeterminate powers
4.7 Newton’s Method
- Procedure for Newton’s Method
- Applying Newton’s Method
- Convergence of Newton’s Method
- But things can go wrong
4.8 Antiderivatives
- Finding antiderivatives, arbitrary constant
- Initial Value Problems and Differential Equations
- Antiderivatives and Motion
- Indefinite Integrals
Ch. 5 Integration
- Idea: compute quantities by breaking into small pieces
and adding the contributions together
5.1 Estimating Finite Sums
- Area: approximating with rectangles, upper/lower sums,
midpoint rule
- Distance traveled knowing the velocity (integrating a DE
to find the model: position=f(times))
- Displacement versus distance traveled
- Average value of a nonnegative function
- Summary: total change as a sum of small changes,
averages
5.2 Sigma notation and limits of finite sums
- Notation, examples
- Algebra rules for finite sums
- Limits of finite sums
- Riemann sums
5.3 The Definite Integral
- Limits of Riemann sums
- Notation and existence of the definite integral
- Theorem: continuous + finite and closedf interval =>
integrable
- Integrable and non-integrable functions: examples,
integrable => bounded
- Properties of definite integrals: rules
- Area under the graph of a nonnegative function
- Average value of a continuous function
5.4 The Fundamental Theorem of Calculus
- Mean Value Theorem for Integrals
- FTC Part 1: Existence of antiderivatives
- FTC Part 2: The Evaluation Theorem
- Total area: using antiderivatives, canceling areas
5.5 Indefinite Integrals and the Substitution Rule
- Indefinite Integrals
- The power rule in integral form
- Substitution: running the Chain Rule backwards
- The integrals of sin^2(x) and cos^2(x)
5.6 Substitution and Areas between Curves
- Substitution formula
- Definite integrals of symmetric functions
- Areas between curves
- Integration with respect to y
- Combining integrals with formulas from geometry