1. Dr. Caulk FPFSC 145 scaulk@regis.edu University Calculus: Early Transcendentals, Second Edition By Hass, Weir, Thomas

2. Grade Breakdown First exam15% Second exam15% Third exam15% Final Exam15% Traditional Homework20% Online Homework10% Classwork10%

3. 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.

4. Course Breakdown/Overview 1. Limits and Continuity 2. Differentiation 3. Applications of Differentiation 4. Integration

5. 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 mid- point, say B 1 of AB. Now to move to B 1 it must first reach the mid-point B 2 of AB 1. Continue this argument to see that A must move through an infinite number of distances and so cannot move.

6. 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.

7. 16 th and 17 th 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.

8. 17 th 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

9. 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.

10. 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.

11. 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.

12. Sources for History http://www.gap- system.org/~history/HistTopics/The_rise_of_cal culus.html http://www.gap- system.org/~history/HistTopics/The_rise_of_cal culus.html http://www.mathteaching.net/math-education/a- brief-history-of-calculus http://www.mathteaching.net/math-education/a- brief-history-of-calculus http://www.saintjoe.edu/~karend/m441/Cauchy.h tml#Augustin-Louis%20Cauchy http://www.saintjoe.edu/~karend/m441/Cauchy.h tml#Augustin-Louis%20Cauchy

13. 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)

14. 1.1 continued Linear Functions Slope of a line Point-slope form: y-y 1 = m(x-x 1 ) Slope-intercept form: y=mx+b Horizontal: y=b Vertical lines: x=a Parallel and perpendicular lines Power functions Polynomials

15. 1.1 continued Trigonometric Functions Exponential Functions Logarithmic Functions

16. 1.2: Combining Functions Sums, differences, products, quotients Composition of Functions Vertical and Horizontal Shifts Scaling and Reflecting

17. 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

18. 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.

19. 1.5: Exponential Functions Rules for exponents – see p. 36. Basic shape of graphs of a x and e x

20. 1.6: Inverse Functions and Logarithms One-to-one Horizontal line test Inverse function Basic shape of log a x and log e x Algebraic properties of the natural logarithm – see p. 43 Inverse sine and cosine.

21. Basic Ideas of Calculus Limits Tangent Line Problem Area Problem