* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Math121 Lecture 1
Survey
Document related concepts
Law of large numbers wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Line (geometry) wikipedia , lookup
Location arithmetic wikipedia , lookup
Infinitesimal wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Positional notation wikipedia , lookup
System of polynomial equations wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Surreal number wikipedia , lookup
Non-standard analysis wikipedia , lookup
Large numbers wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Transcript
Math121 Lecture 1 Lecturer: Dr. Robert C. Busby Office: Phone : Korman 266 215-895-1957 [email protected] Email: Course Web Site: http://www.mcs.drexel.edu/classes/Calculus/MATH121_Fall02/ (Links are case sensitive) Structure of the Course 1. 3 lectures a week, 2 recitations 2. Book – Anton Calculus Seventh Edition 3. Software Graphing Advantage Graph2D Graph2D We are all familiar with the Rational Number system – all whole numbers and fractions, both positive and negative: For example 1, − 4, 1 , − 3 , 234, − 236 1, 23.14 2 46 11 2 Are all examples of rational numbers. We can add, subtract, multiply and divide these numbers (except that we can’t divide by 0), and we have all the rules we learned about arithmetic. These include: a +b =b +a ab = ba (Commutative Laws) (a +b) + c =a + (b + c) (ab)c = a(bc) (Associative Laws) a(b + c) = ab + ac (Distributive Law) We can visualize rational numbers geometrically by plotting them on a straight line, which we then refer to as a number line. We begin by fixing a point on the line to represent the number 0. Then we choose a unit length, and represent 1 by the point lying to the right an amount equal to that length. Then every other number is represented by a proportional length, with negative numbers plotted to the left of 0. The Number Line has many points that correspond to rational numbers −2 −1 0 1 2 1 5 4 3 2 7 4 2 Plotting all possible rational numbers on the number line actually leaves many holes. −2 −1 0 1 2 1 5 4 3 2 7 4 2 To see this, we recall a theorem from geometry. Therefore c2 c b a a2 b2 2 1 1 2 1 0 1 2 The square root of 2 cannot be a fraction. Suppose it was p over q. We can assume that p and q are not both even, by cancellation. Then: p 2 = 2 ⇒ p2 = 2q2 so p is even, say p = 2r. q2 4r 2 = 2q2 so 2r 2 = q2 so q is even. We can construct the set of numbers corresponding to all possible lengths, both rational and irrational. This set is called the real numbers, and we can do arithmetic with them as with rational numbers, with all the familiar rules being true. These numbers are characterized by the fact that their decimal form has no repeated pattern (or pattern of any kind). π =3.14159... We can therefore identify any straight line with the real numbers by choosing an origin and a unit.The number becomes the label or address of the corresponding point, and is called its ‘rectangular coordinate’. x −1 0 1 2 3π 4 We often use a ‘variable’ name (x, y, t, s, u) as the label or address some unspecified point on the line, that is an arbitrary real number. This idea also lets us assign labels, or addresses to arbitrary points in a plane. We choose an origin through which we draw a horizontal and a vertical line. We choose unit lengths on these lines (normally equal, but not necessarily). Then every point P in the plane is uniquely identified by its horizontal distance x from the origin and its vertical distance y from the origin. These numbers are called the Rectangular coordinates of P, and we write (x, y) instead of P when we want to mention them explicitly. y 3 2 (2,2) 1 -3 -2 -1 0 -1 -2 (-2,-3) -3 (3,1) 1 (0,0) 2 3 x (3,0) (x2 ,y2 ) y2 − y1 d (x1 ,y1) x2 − x1 d= ( x2 − x1 ) + ( y2 − y1 ) 2 2 3 d= (-1.5, 2.5) 2 1 -3 -2 = = 20.25+12.25 = 32.5 ≈ 5.7 1 -1 ( 3−(−1.5) ) +( (−1) −2.5) 2 2 ( 4.5) + ( −3.5) 2 2 3 (0,0) -1 -2 -3 (3, -1) 2 Find a condition on x and y that characterizes the points that lie on a circle of center (3,2) and radius 1. This is equivalent with the algebraic condition 1= ( x − 3) + ( y − 2 ) 2 2 or (x, y) 2 + y − 2 2 =1 x − 3 ( ) ( ) 1 (3, 2) Trigonometric Functions Sin(θ) = y/r Cos(θ) = x/r r θ x y Tan(θ) = y/x Cot(θ) = x/y Sec(θ) = r/x Csc(θ) = r/y Trigonometric Identities r θ y x2+ y2 = r2 x Sin 2(θ) + Cos2(θ) = 1 Sin(A + B) = Sin(A)Cos(B) + Cos(A)Sin(B) Cos(A + B) = Cos(A)Cos(B) − Sin(A)Sin(B)