Science Teaching 552

Calculus on a Computer

Course Outline

 

Syllabus

 

1        Course Logistics

•         Homework – each session

–        At http://infohost.nmt.edu/~wdstone/

•         Click on ST 552

•         No tests J

•         Contact information

–        wdstone@nmt.edu

–        Office: 505-835-5786

–        Fax: 505-835-5366

–        Dept. of Math, NM Tech, Socorro, NM 87801

•         Review of some topics from Algebra

•         The Number Line

•         Distance

•                     Absolute value

•         Circles

•         Lines

•         Review of some topics from Trigonometry

•         Radians

•         The Unit circle

•         The basic trig functions

•         More on lines

•         Slope-Intercept

•         Point-Slope

•                     Parallel & Perpendicular lines

•                     Functions

•                     Definition

•                                 Composition

•                     Polynomials

•                     Rational functions

•                     Graphing

•                     By hand

•                     With MAPLE

2                    Inverse functions, logarithms & exponentials

•                     Inverses

•                     Definition

•                     Notation

•                     Graphs

•                     Restricted range

–                                Trig functions

•                                 Calculating inverses

•                     Exponentials

•                                 Definition

–                                Negative powers

–                                Fractional powers

•                                 Rules of exponentials

–                                Product, quotient, powers

•                                 Graphs

•                                 Applications

•                                 Logarithms

•                                 Definition

•                                 Rules

–                    From rules for exponents

–                    Different bases

•                     Graphs

•                     Applications

•                     Story problems!

•                     Compound interest

•                     Radioactive decay

–                    C14 dating

•                     Heating & Cooling       

•                     Population growth

•                     Drug dosing

3                    Derivatives

•                     Slopes

•                     Lines

•                     Applications

–                    Hills

–                    Velocity

•                     Instantaneous?

•                     Concepts

•                     Rate of Change!

•                     Slope

•                     Velocity

•                     Acceleration

•                     Population growth

•                     Concentration change

•                     Tangents

•                     Limit of secants

•                     Examples

•                     Definition

•                     By hand

•                     With MAPLE

•                      “Zooming”

•                     Examples & pictures

•                     Linear approximation

•                     Applications

•                     Square roots

•                     Population growth

•                     Units

4                    Rates of Change

•                     Velocity & Acceleration

–                    Average vs. instantaneous

•                     Speed vs. velocity

•                     Graphs

•                     Falling bodies

•                     Other rates

•                     Areas & volumes

•                     Flow

•                     Conservation laws

•                     Story problems

•                     Falling body problems

•                     Population problems

•                     Velocity problems

•                     Linear approximation

•                     Function approximation

•                     Newton’s method

–                    Geometric

–                    Algebraic

•                     More story problems

•                     Finding roots

•                     Heating problems

•                     Volume & surface area problems

5          The Chain Rule

•                     Composition of functions

•                     Definition

•                     Notation

•                     Domains & ranges

•                     The Chain Rule

•                     Sketch of the proof

•                     Newton’s notation

•                     Leibniz’ notation

•                     Examples

•                     Extended version

•                     Related rate story problems

–                    Similar triangles

–                    Angles

–                  Volumes

6          Implicit Differentiation & Parametric Equations

•                     Derivatives of exponentials

•                     Compound interest

•                     Derivative of exponentials

•                     exp(x) & ln(x)

•                     Implicit equations

•                     Definition

•                     Examples

•                     Implicit differentiation

•                     Examples

•                     Comparison with other methods

•                     Derivatives of inverse functions

•                     Derivation

•                     ln(x) vs. log10(x)

•                     Inverse trig functions

•                     Other examples

•                     Parametric equations

•                     Definition

•                     In MAPLE

•                     Examples

•                     Parametric differentiation

•                     From the chain rule

•                     Examples

•                     Comparing with implicit

7          Max-Min problems

•                     Critical points

•                     Definition

–                    Two types

•                     Finding critical points

•                     Extreme values

•                     First derivative test

•                     Bounded domains

•                     Local minima/maxima

•                     What do we have to check?

–                    Critical points

•                      

 

 

•                      

 

–                    Endpoints

•                     Global minima/maxima

•                     What do we have to check?

–                    Compare local extreme values

•                     Story problems

•                     Geometry

•                     Material

•                     Costs/profits

•                     Important points:

–                    Domain of realistic solutions

–                    Check the endpoints

8          More Max-Min problems

•                     Second derivative test

•                     Meaning of the second derivative

•                     Examples

•                     Second derivative test

•                     Examples

•                     Cautionary examples

•                     Story problems

•                     Pipe around a corner

•                     Viewing angles

•                     Flow

•                     Harvesting

•                     Profits

9          Integrals – Areas as Sums

•                     Areas as limits

•                     Rectangles

–                    Max vs. min

–                    Left endpoint vs. right endpoint

–                    Midpoint

•                     Trapezoids

•                     Summation notation

•                     Σ notation

•                     Sequences

•                     Examples

•                     Some specific sums

 

 

•                      

 

 

•                      

 

 

•                      

 

 

•                     Examples

 

•                      

 

 

•                      

 

•         More examples

10        Anti-derivatives & the Fundamental Theorem of Calculus

•         Area under a curve as a function

•         Simple examples

•         (a line)

•         Different starting points

•         General picture

•         The Fundamental Theorem

•         Derivative of the area function

•         Examples

•         Anti-derivatives

•         Non-unique

–        Geometric meaning

•         By hand

–        Polynomials

–        1/x?

•         Using MAPLE

11        More on Anti-derivatives

•         Geometric interpretation

•         Area

•         Function vs. slope

•         Graphically reconstructing a function from its derivative

•         Examples

•         Polynomials

•         Trig functions

•         General

•         Separable differential equations

•         A clever abuse of the notation

•         Linear problems

•         Homogeneous

•         Story problems

•         Growth & decay

•         Cooling

•         Mortgage

•         Population models

12        Applications of the Definite Integral

•         Area

–        Cut it up, add it back together

•         Volumes by slices

–        Cut it up, add it back together

•         Work

–        Cut it up, add it back together

•         Velocity to displacement

–        Cut it up, add it back together

•         Hydrostatic pressure

–        Cut it up, add it back together

•         Volume by shells

–        Cut it up, add it back together

•         Formulζ from high school physics

•         Fine if everything is constant, but if its not:

–        Cut it up, add it back together

13        More Applications

•         Arc length

•         Pythagorean formula

•         A cautionary tale

•         Various forms

–        dx

–        dy

–        dt

•         Examples

•         Surface area

–        (Solids of rotation)

•         Geometry

•         The formula

•         Examples

14        Integration techniques

•         Substitution

•         The Chain rule

•         Examples

•         Integration by parts

•         Product rule

•         Applications

•         Examples

•         Trig substitutions

•         Trig identities

•         Examples

•         Partial fractions

•         Algebraic manipulations

•         The basic form

•         The ‘rule of thumb’

•         Examples

15        Infinite Series

•         Sequences

–        Convergence and divergence

•         Definition

•         Convergence

•         Examples

•         Series

•         Convergence & divergence

–        Sequence of partial sums

•         General properties

–        linearity

•         Examples

–        Geometric series

–        Telescoping

•         Convergence tests

•         Divergence test

•         Comparison test

–        Limit comparison

•         Integral test

•         Root test

•         Ratio test

•         Alternating series test

–        Absolute & conditional convergence

16        Power Series & Taylor Series

•         Power series

•         Convergence

•         Radius of convergence

•         Term by term differentiation & integration

•         Taylor Series

•         Taylor’s theorem

•         Uniqueness

•         Remainder theorem

•         Examples

•           ex

•         sin(x)

•         cos(x)

•          1/(1-x)

17        Overview