Dr. CaulkFPFSC 145

scaulk@regis.eduUniversity Calculus: Early Transcendentals, Second

EditionBy Hass, Weir, Thomas

Grade BreakdownFirst exam 15%

Second exam 15%

Third exam 15%

Final Exam 15%

Traditional Homework 20%

Online Homework 10%

Classwork 10%

Some StandardsHave a meaningful positive experience.Exercise and improve quantitative and logical

reasoning skills.Exercise and improve ability to communicate

mathematical ideas both orally and in writing.

Course Breakdown/Overview1. Limits and Continuity2. Differentiation3. Applications of Differentiation4. Integration

Greeks and CalculusGreeks equated numbers to ratios of integers,

so to them, the number line had “holes” in it. They got around this problem by using lengths, areas and volumes in addition to numbers.

Zeno (450 BC): If a body moves from A to B then before it reaches B it passes through the mid-point, say B1 of AB. Now to move to B1 it must first reach the mid-point B2 of AB1 . Continue this argument to see that A must move through an infinite number of distances and so cannot move.

Greeks continued…

Exodus (370BC) Method of Exhaustion – calculate areas by thinking of them as an infinite collection of shapes that are easy to compute.

Archimedes (225BC) Used an infinite sequence to compute the area of a segment of parabola. Also used method of exhaustion to approximate the area of a circle.

16th and 17th Century ContributionsKepler found area of ellipse by thinking of

areas as sums of lines.

Roberval thought of the area between a curve and a line as being made up of an infinite number of infinitely narrow rectangular strips.

Fermat generalized the area of a parabola and hyperbola, and computed maxima and minima.

17th Century ContributionsBarrow used a method of tangents to a curve

where a tangent is the limit of a line segment whose endpoints move toward each other.

Barrow's differential triangle

NewtonNewton wrote about fluxions in 1666. He

thought of a particle tracing a curve with a horizontal velocity (x’) and a vertical velocity (y’) y’/x’ was the tangent where the curve was horizontal.

He also tackled the inverse problem (antidifferentiation) and stated the Fundamental Theorem of Calculus.

Newton thought of variables changing with time.

LeibnizLeibniz thought of variables x and y as

ranging over sequences of infinitely close values and introduced dx and dy notation as differences between successive values in these sequences.

Integral calculus and the integral operator were developed by Leibniz.

Rigor Introduced into CalculusCauchy(1821) Rigorous definition of limit in

class notes for Course on Analysis:" When the values successively attributed to a

particular  variable approach indefinitely a fixed value so as to  differ from it by as little as one wishes, this latter value is called the limit of the others. "

Weierstrass introduced absolute values and epsilons and deltas in the definition we use today.

Sources for Historyhttp://www.gap-system.org/~history/

HistTopics/The_rise_of_calculus.htmlhttp://www.mathteaching.net/math-

education/a-brief-history-of-calculushttp://www.saintjoe.edu/~karend/m441/

Cauchy.html#Augustin-Louis%20Cauchy

1.1 Functions and Their GraphsDomain and rangeRepresentations: Formula, Table of Values,

GraphVertical Line Test for FunctionsPiecewise defined functionsA function y=f(x) is an

even function of x if f(-x)=f(x) odd function of x if f(-x)=-f(x), for every x in the domain of f. (p.6)

1.1 continuedLinear Functions

Slope of a line Point-slope form: y-y1 = m(x-x1)Slope-intercept form: y=mx+bHorizontal: y=bVertical lines: x=a

Parallel and perpendicular linesPower functionsPolynomials

1.1 continuedTrigonometric FunctionsExponential FunctionsLogarithmic Functions

1.2: Combining FunctionsSums, differences, products, quotientsComposition of FunctionsVertical and Horizontal ShiftsScaling and Reflecting

1.3: Trigonometric FunctionsSix basic trig functionsValues at standard angles – see p. 23The graphs of sine, cosine, tangentThe unit circleBe able to look up and use identities

1.4: Graphing with Calculators and ComputersUse whatever calculator you already have for

homework, etc. No calculators for tests.Maple on computers on campus.

1.5: Exponential FunctionsRules for exponents – see p. 36.Basic shape of graphs of ax and ex

1.6: Inverse Functions and LogarithmsOne-to-oneHorizontal line testInverse functionBasic shape of logax and logexAlgebraic properties of the natural logarithm –

see p. 43Inverse sine and cosine.

Basic Ideas of CalculusLimitsTangent Line ProblemArea Problem