Transcript MT360A: Calculus I Fall 2011

Dr. Caulk
FPFSC 145
[email protected]
University Calculus: Early Transcendentals, Second Edition
By Hass, Weir, Thomas
Grade Breakdown
First exam
15%
Second exam
15%
Third exam
15%
Final Exam
15%
Traditional Homework
20%
Online Homework
10%
Classwork
10%
Some Standards
 Have 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/Overview
1. Limits and Continuity
2. Differentiation
3. Applications of Differentiation
4. Integration
Greeks and Calculus
 Greeks 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 midpoint, 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.
th
16
th
17
and
Century
Contributions
 Kepler 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.
th
17
Century Contributions
 Barrow 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
Newton
 Newton 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.
Leibniz
 Leibniz 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 Calculus
 Cauchy(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 History
 http://www.gap-
system.org/~history/HistTopics/The_rise_of_cal
culus.html
 http://www.mathteaching.net/math-education/abrief-history-of-calculus
 http://www.saintjoe.edu/~karend/m441/Cauchy.h
tml#Augustin-Louis%20Cauchy
1.1 Functions and Their Graphs
 Domain and range
 Representations: Formula, Table of Values, Graph
 Vertical Line Test for Functions
 Piecewise defined functions
 A 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 continued
 Linear Functions
 Slope of a line
 Point-slope form: y-y1 = m(x-x1)
 Slope-intercept form: y=mx+b
 Horizontal: y=b
 Vertical lines: x=a
 Parallel and perpendicular lines
 Power functions
 Polynomials
1.1 continued
 Trigonometric Functions
 Exponential Functions
 Logarithmic Functions
1.2: Combining Functions
 Sums, differences, products, quotients
 Composition of Functions
 Vertical and Horizontal Shifts
 Scaling and Reflecting
1.3: Trigonometric Functions
 Six basic trig functions
 Values at standard angles – see p. 23
 The graphs of sine, cosine, tangent
 The unit circle
 Be able to look up and use identities
1.4: Graphing with Calculators
and Computers
 Use whatever calculator you already have for
homework, etc.
 No calculators for tests.
 Maple on computers on campus.
1.5: Exponential Functions
 Rules for exponents – see p. 36.
 Basic shape of graphs of ax and ex
1.6: Inverse Functions and
Logarithms
 One-to-one
 Horizontal line test
 Inverse function
 Basic shape of logax and logex
 Algebraic properties of the natural logarithm –
see p. 43
 Inverse sine and cosine.
Basic Ideas of Calculus
 Limits
 Tangent Line Problem
 Area Problem