Introduction to Mat 146
(Calculus II)
- Goals & Motivation
- How the course is organized & expectations
- Brief recall of Calculus I
- Student background information & feed-back
“LearningCalculus”
My Goals:
1) Help you acquire math knowledge (what for? …)
2) Help you develop your learning skills (what for? …)
Your Objectives:
- … and a good grade!
Resources for Calc II
- Instructor: Lucian Ionescu
(Office hours, , Mat146 Web site)
- Text: Thomas’ Calculus 11EE
Content:
6.Applications of integration
7. Log, exp & applications
8.Techniques of integration
9Differential equations (more gen. Int.)
10Infinite sequences and series (extend poly)
- Calculator with a Computer Algebra System (CAS): TI-89/92 (Symbolic differentiation, integration, solving D.E., …)
- Syllabus
1) Lecture – recitation format
2) Learning cycle
3) List of homework assignments – due W
4) Outlines, participation, attendence
5) Evaluation / grading scale
“Calculus: 1,2,3…”
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.
The Learning Cycle
- What is “learning”? Acquisition of
- Knowledge
- Learning skills
- How “to learn”? (Repetition is the key)
1. BEFORE class, skim through the assigned section (1/2 hour)
(Identify “keywords”).
2. Come to class and:
- Answer questions from the previous section (review);
- 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, and write an OUTLINE
(Concepts & steps of procedures);
4. Attempt all the homework problems (C):
a) First, without the help of the book;
b) If it doesn’t work, use the book (go back to 2. above);
c) Write a list of questions to ask (group/instructor).
- Impact on grade: 1-4 leads to A!
Solving a Problem:
Thinking and Computing
I) High level mathematical language: “Modeling the real system”
- Concepts and relations: derivative, integral, etc. & theorems;
- Models: logistic model, etc.
- Methods to solve the models (“algorithms”): D.E., etc.
II) Mathematical code:
- Formulas and algebraic manipulations (rules to differentiate)
- Computations (evaluate a function, etc.)
- Executing the algorithms (solving a quadratic eq.)
Human’s job: (I)
Computer’s task: (II)
Your Concern (Task):
Translate a PROBLEM (I) into computer’s task (II).
Conclusions: what to learn?
- “The Theory”:
- Concepts and relations;
- Methods and Procedures
- How to “setup equations” (tasks for the computer)
- How to use the calculator
- How to interpret the results.
Example: “Volume by slicing”
(What to learn?)
- How to understand “Volume” (the concept):
1) Intuitively (Other sciences, own experience etc.)
2) Mathematically (measure, # of cubes -> )
- How to compute “Volume” (the math procedure):
- Input: type of data (description of the solid),
- Output: volume (positive number w. meaning!),
- Algorithm: adapting the input to the appropriate formula & performing the integration using the calculator (or by hand, to develop strong computational abilities).
Calculus I Recall
- Derivative & rate of change;
- Anti-derivatives = indefinite integral
- Integral & total change
- Riemann integral = definite integral
- FTC
- Part I: existence of antiderivatives;
- Part II: computing definite integrals.
- Interpretation: differentiation and integration are operations inverse to one another
Definite or Indefinite Integral
(What’s the difference anyway?)
I) Definite integral provides an antiderivative
(“Definite => indefinite”)
II) Antiderivatives allow to compute definite integrals
(“Indefinite => definite”)
Math 146 – Feedback no.1Name:______
1. If f(x)=x e3x, compute f ’(x)=
2. Find the local linearization of f(x)=arctan(x) at x=1.
3) Find the indefinite integral of x3.
4) When did you take Calc I?
5) What are YOUR goals in Calc II?
- Comments and suggestions:
- I wish …
Ch. 6 Applications of definite integrals
Recall:
- Riemann sums and integration = “Total change”
- FTC and total change from rate of change.
6.1 Volume by slicing
(ppt slides & examples)
Summary
- procedure “volume by slicing”
- input= description of the solid,
- output=volume
- idea: dimensional reduction with Cartesian coordinates
- algorithm / formula = integral of cross-section area (plane regions)
- Cavalieri’s principle (consequence)
- Solids of revolution (special description of solid => special cross-section):
1) disk method (cross-sections = disks)
2) washer method (c-s = washers)
- Examples / more practice
6.2 Volumes by cylindrical shells
- Same as “Volume by slicing”, except using polar coordinates for solids of revolution
Summary
- procedure: “volume by slicing”, input: description of the solid, output: volume
- idea: dimensional reduction with polar coordinates
- Cross-sections by constant coordinate surfaces: infinite cylinders
- algorithm / formula = integral of cross-section area (cylinders)
- Review & questions, discuss homework
- New material: keywords, ppt presentation (lecture), summary
- Peer concept check
6.3 Lengths of plane curves
- Idea: approximate + Pythagora+limit
- Parametric curves
- Justifying the formula: Pythagora => length formula
- Example: input, formula / TI-89 => output
- Graph of a function:
- Justification: special case of parameterization x=t, y=f(x)=f(t)
- Example
- Differential formula: L= ds, ds=(dx2+dy2)½
- Summary: s, ds, parameterization, graph (special parameterization)
6.4 Moments and the Center of Mass
- Center of Mass
- Torque of a mass: mgx; moment: mx
- Main property of CM: equilibrium = cancellation of torques / moments
- Formula;
- Interpretation: coordinate of CM = average coordinates (with weights)
- 1dim objects: Wires and Thin rods
- 2dim objects: Thin flat plate (1-integrals)
- Examples – practice
6.5 Areas of surfaces of revolution and the Theorem of Pappus
- Surface area
- Dimensional reduction with Cartesian coordinates
- Surface of revolution: cross-sections=circles
- “Same (type) of formula”: the differential form L(x)ds,
- Cross-section circumference: L(x)=2 r(x)
- Differential arc length:
1) graph: ds=(1+f’2)½ ,
2) parameterized curve ds2=dx2+dy2
- Theorem of Pappus
1) Volume of Solid of revolution: A(cross-section) x L(Centre of Mass Circle)
justification: Average circumference=CM Circle
2) Area of Surface of Revolution= L(cross-section) x L(Centre of Mass Circle)
justification: same, 1-dimension lower.
6.6 Work
Work
- Work done by a constant force
- Work done by a variable force in 1-dimension
Examples
- Elastic force & Hooks law for Springs (F=-kx)
- Gravity & Pumping liquids from containers (F=mg)
6.7 Fluid Pressures and Forces
Pressure
- Pressure = Force / unit of surface; scalar: same in all directions at a point
- The Pressure-Depth Equation: p=wh
Force
- Constant-depth formula for the fluid force: F=pA
- Variable-depth formula: F=p dA, vertical plate: dA=width x dh
- Fluid forces and centroids: F=w hc A
Ch. 7 Integrals and Transcendental Funcations
- A precise introduction of exp, ln and hyperbolic functions (analog of cos & sin)
7.1 The Logarithm define as an Integral
The natural logarithm function
- Def. ln x=1/t dt, a few values
- Number e
- The derivative of ln x and of ln|x|
- Graph and range of ln x
- Properties of ln with proofs
- The integral of ln u (u<0 and u>0)
- Example 1
The inverse of ln(x):
- The natural exponential
- The derivative, integral, example 2 (IVP)
- Properties: laws of exponents
General exponential function
- Def. a^x, log_a(x), derivatives and integral
- Properties
- Example 3
7.2 Exponential Growth and Decay
The law of exponential change
- Def.
- The DE model
Applications
- Unlimited population growth
- Continuous compounded interest: k times/year and continuous limit
- Radioactivity
- Heat transfer and Newton’s Law of Cooling
7.3 Relative rates of growth
- Which functions grow faster? Poly, exp, log …
- Comparing growth rates:
1) Asymptotically (“at infinity”)
2) At a point: rates of change and relative rates of change
1) Order of magnitude: lim f(x)/g(x)= finite, 0, infinite
- Grows at the “same rate”, “slower than”, “faster than”
2) Rates if change of functions at a point:
- Absolute: y’(a);
- Relative: y’(a)/y(a);
- Example: Population growth, bank account amount
7.4 Hyperbolic functions
- Abstract: analogs of trigonometric functions
- Even and odd parts of the exponential
- Def. hyperbolic functions sinh, cosh,
- Main identity, geometric interpretation: circle & (equilateral) hyperbola
- Derivatives and integrals
- Derived hyperbolic functions: tanh, coth, sech, csch etc.
- Inverse hyperbolic functions: def., derivatives and integrals (use calculator);
Ch. 8 Techniques of Integration
- Recall FTC
8.1 Basic Integration Formulas
- Substitution rule
- Examples
8.2 Integration by parts
- Product rule in integral form
- Integration by parts formula
- Evaluating definite integrals by parts
8.3 Integration of rational functions by partial fractions
- Example of a decomposition
- General description of the Method
- Reduction of degree of numerator
- Examples
- Heaviside cover-up method for linear factors – Example 6
8.4 Trigonometric integrals
- Products of powers of sines and cosines
- Eliminating square roots
- Integrals of powers of tan x and sec x
- Products of sines and cosines (Example 7)
8.5 Trigonometric substitutions
- Three basic substitutions
- Examples
8.6 Integral Tables and Computer Algebra Systems
- Integral tables
- Reduction formulas
- Non-elementary integrals
- Integration with a CAS
8.7 Numerical Integration
- Trapezoidal approximations – the trapezoidal rule
- Error estimate for the trapezoidal rule
- Simpson’s rule: approximating using parabolas
- Error estimate for Simpson’s rule
8.8 Improper integrals
- Infinite limits of integration
- Integrands with vertical asymptotes
- Tests for convergence and divergence
- Direct comparison test
- Limit comparison test
Ch. 9 Further applications of integration
- From integral (definite/indefinite) to more general DE
9.1 Slope fields and separable differential equations
- General 1st ODE
- Slope fields: viewing solution curves – use calculator
- Separate equations
- Torricelli’s law – Ex.5
9.2 First-order linear DE
- The growth/decay equation, having a linear form
- Standard form, Ex.1
- Solving linear equations
- RL Circuits
- Mixture problems
9.3 Euler’s Method
- Euler’ method
- Improved Euler’s method
9.4 Graphical solutions of autonomous DE
- Equilibrium values and phase lines
- Stable and unstable equilibria
- Population growth in a limited environment
9.5 Applications of 1st ODE
- Resistance proportional to velocity
- Modeling population growth
- relative growth rate
- carrying capacity
- Orthogonal trajectories
Ch. 11 Infinite sequences and series
11.1 Sequences
- Infinite sequences
- Convergence and divergence
- Diverges to infinity
- Calculating limits of sequences
- The sandwitch theorem for sequences
- Using l’Hospital Rule
- Commonly occurring limits
- Recursive definitions
- Bounded non-decreasing sequences
- The non-decreasing Theorem
11.2 Infinite series
- Geometric series
- Divergent series
- The nth-term test for divergence
- Combining series
- Adding or deleting terms
- Reindexing
11.3 The integral test
- Non-decreasing partial sums
- The integral test
11.4 Comparison tests
- The comparison test
- The limit comparison test
11.5 The ratio and root test
- The ratio test, - The root test
11.6 Alternating series, absolute and conditional convergence
- Alternating series, harmonic series
- The Alternating Series Estimation Theorem
- Absolute and conditional convergence
- Absolute convergent
- Conditionally convergent
- The absolute convergence test
- Rearranging series
11.7 Power series
- Power series and convergence
- The convergence theorem for power series
- The radius of convergence of a power series
- Term-by-term differentiation
- Term-by-term differentiation theorem
- Term-by-term integration theorem
- Multiplication of power series
11.8 Taylor and MacLaurin series
- Series representations
- Taylor and MacLaurin Series
- Taylor polynomials
- Taylor polynomial of order n
11.9 Convergence of Taylor Series; error estimates
- Taylor Theorem / Taylor’s formula
- Estimating the remainder
- Truncation error
- Combining Taylor series
- Euler’s identity
11.10 Applications of power series
- The binomial series for powers and roots
- Power series solutions of DE
- Evaluating nonelementary integrals
- Arctangents
- Evaluating indeterminate forms
11.11 Fourier Series
- Fourier series, finding coefficients
- Convergence of Fourier Series