Math 1448C Mathematics Tuesday 13 October 2020 Instructor: dr Adam - PowerPoint PPT Presentation

Math 1448C “ Mathematics” Tuesday 13 October 2020 Instructor: dr Adam Abrams

Course overview Absolute values, polynomials Logarithms and exponents Matrices and systems of linear equations Calculus Recurrence relations Limits of sequences and limits of functions Derivatives and integrals Minimum and maximum values

Course format Lecture (Wyk ł ad) - dr hab. Jacek Serafin 11:15 - 13:00 every Tuesday 
 Problem session ( Ć wiczenia) 7:30 - 9:00 every Tuesday for group code Z03-87b 9:15 - 11:00 every Tuesday for group code Z03-87a 
 30 7.5 Grades: points from lecture (final exam) and points from active participation in problem sessions. See http://prac.im.pwr.edu.pl/~serafin/dydaktyka.html

Draw a cube These are all correct! If multiple people draw or talk about a cube, we need to be sure we are all thinking of the same thing.

⃗ ⃗ ⃗ Vocabulary Variable - a symbol (letter) used to represent an unknown number, vector, or other mathematical object x y z n v A Examples: Expression - a combination of symbols not including =, ≤ , <, etc. 3 x 2 + 5( x − 1) 3 x 2 cos(2 x ) u + v Examples: expression 1 = expression 2 Equation - a statement of the form “ ”. Function (or mapping) - a rule for changing inputs to outputs Coefficient - a constant that is multiplied by some power of a variable 8 x 3 + x 2 + 5 x x 3 8 Example: In “ ”, the “coefficient of ” is , the x 2 1 5 x “coefficient of ” is , and the “coefficient of ” is .

maybe Algebra review 6 × a 6 ⋅ a 6 a a “ ” and “ ” and “ ” all mean six times . 6( a + b ) 6 a + 6 b can be re-written as . 3 x − 12 3( x − 4) can be re-written as . This is “factoring”. ( x + 7)( y + 2) xy + 2 x + 7 y + 14 can be expanded to . x 2 + 14 x + 49 ( x + 7) 2 can be expanded to . 
 a 2 + 2 ab + b 2 ( a + b ) 2 In general, expands to . x 2 + 14 x + 49 ( x + 7) 2 can be factored as .

Be careful! a 2 × b 2 ( a × b ) 2 can be re-written as . a 2 + b 2 ( a + b ) 2 can not be re-written as . Try it with actual numbers: 
 (2 + 3) 2 = 5 2 = 25 2 2 + 3 2 = 4 + 9 = 13 , but . Testing specific numbers can only show you when a rule is false . 
 It cannot guarantee that a rule is correct because you might pick (0 + 0) 2 = 0 = 0 2 + 0 2 numbers where it accidentally works, like . a + b ≠ a + b sin( a ⋅ b ) ≠ sin( a ) ⋅ sin( b )

Polls 🚬 🚬 ✅ poles Poles polls

Types of numbers 0, 1, 2, 3, 4, . . . Natural numbers: 1, 2, 3, 4, . . . In some books, only . . . , − 3, − 2, − 1, 0, 1, 2, 3, 4, . . . Integers : Rational numbers are all the numbers that can be written as one 1 2 , − 2 3 , 1.5, 8, 0, − 5 integer divided by another. Examples: 4 Real numbers are all the values on a number line. Examples: − 9 − 3 0 1 4.8 π − 2.718... 3 2 2


 
 Absolute value Algebra idea: make numbers positive Geometry idea: measure distance | x | x We write for the absolute value of . − 9 2 is 9 | 5 | is 5 | − 3 | is 3 | 37.2 | is 37.2 Examples: 2


 
 Absolute value Algebra idea: make numbers positive Geometry idea: measure distance | x | x We write for the absolute value of . − 9 2 is 9 | 5 | is 5 | − 3 | is 3 | 37.2 | is 37.2 Examples: 2 | x | = { x if x ≥ 0, Definition, version 1: − x if x < 0.

Absolute value Algebra idea: make numbers positive Geometry idea: measure distance | x | x We write for the absolute value of . | x | 0 x Definition, version 2: is the distance between and . distance 4.8 − 9 − 3 0 1 4.8 π − 2.718... 2 3 2

Absolute value Algebra idea: make numbers positive Geometry idea: measure distance | x | x We write for the absolute value of . | x | 0 x Definition, version 2: is the distance between and . distance 3 − 9 − 3 0 1 4.8 π − 2.718... 2 3 2


 Absolute value | 5 − 3 | How can we think of in terms of distances? 
 5 − 3 First, what does mean? | 5 − x | How can we think of in terms of distances?

Multiplication 5 × 3 What does mean? 5 + 5 + 5 More advanced: no pictures, just . 
 5 × 1 5 × 9.2 7.65 × ( − 12) What does mean? 
 ? ? 
 3 Depending on the context, multiplication can have 
 different meanings or interpretations. This is also true for subtraction.


 
 
 
 Subtraction 5 − 3 What does mean on a number line ? 
 +2 3 5 5 − 3 3 5. Answer: The number describes how to move from to 5 3 3 − 5 To go from to instead, we move left , which is why is negative.


 
 
 
 Subtraction 5 − 3 What does mean on a number line ? 
 +3 +6 +2 − 4 − 2 − 1 3 4 5 5 − 3 3 5. Answer: The number describes how to move from to 5 3 3 − 5 To go from to instead, we move left , which is why is negative.


 
 
 
 Subtraction 5 − 3 What does mean on a number line ? 
 +3 +6 +2 − 4 − 2 − 1 3 4 5 5 − 3 3 5. Answer: The number describes how to move from to 5 3 3 − 5 To go from to instead, we move left , which is why is negative. b − a In general, the number a b describes how to move from to .

Absolute value again | 5 − 3 | How can we think of in terms of distances? 5 − 3 3 5 First, describes how to move from to . The absolute value means we don’t care whether the previous line gives a negative or positive value. Whether you move left or right doesn’t matter, only the distance. | 5 − 3 | 5 3 Answer: is the distance between and . | 5 − x | How can we think of in terms of distances? | 5 − x | 5 x Answer: is the distance between and .

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