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

• Formulæ 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

^• e^x

• sin(x)

• cos(x)

• 1/(1-x)

17 Overview