Math Camp 2005 Instructor: Udi Sommer

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Math Camp 2005Instructor Udi Sommer

• Second Class (Thursday, August 18, 10-1a)
• logarithms
• functions / relations
• limits and continuity
• asymptotes
• differentiability
• well-behaved functions

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Logarithms

• Definition For a positive number n, the
  logarithm of n is the power to which some number
  b (the base of the logarithm) must be raised to
  yield n.
• Thus if bx n, then log b n x (Hagle, 1995, p.
  2)
• log (a X b) log a log b
• log (a / b) log a log b
• log (an) n log a
• log a ax x ln ex x

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Relations / Functions

• A Relation, R, is an association between two or
  more objects (Hagle, 1995)
• The Domain of a relation is the set of all
  possible x values of the relation
• The Range of the relation is the set of all
  possible y values of the relation.
• And thus, the ordered pair (x, y) is an element
  of the relation R.

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Functions

• A Function associates elements of one set with
  elements of another set.
• It provides a rule assigning elements of the
  Range to elements of the Domain.
• A Function is a special type of relation such
  that no two elements of the relation have the
  same x. In other words, if a certain value of y
  is associated with a certain value of x, that
  value of x is not associated with any other value
  of y.Furthermore a certain value of y might or
  might not have a value of x associated with it.
• x ? y (? ? ?)

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Laws of Functions

• ? g y (x) ? g(x) addition and subtraction
• a ? y a(x) for a ? ? multiplication by
  consonant
• ? g y (x) ? g(x) multiplication
• a y (x)a for a ? ? raise to the power of a
• / g y (x) / g(x), provided that g(x) ?
  0 division
• g ? y g (x) composition

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Types of Functions Linear

• Algebraically those functions take the form y
  mx b
• Where m is the slope of the functions line. In
  essence it is a fraction where the numerator is
  the rise or distance on the y-axis, and the
  denominator is the run or distance on the
  x-axis.
• And where b is the y intercept. Or in other
  words, this is the place where the line crosses
  the y-axis.
• more than one independent variable y (DV) b0
  b1x1 b2x2 b3x3 (IVs) ?bixi

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Types of functions

• ii. Constant functions
• iii. Finding the equation of the line w/
  coordinates of a point and a slopey y1 m (x
  - x1)
• iv. Finding the equation of the line w/
  coordinates of two pointsy y1 (y2 y1) /
  (x2 x1) (x x1)
• v. Systems of Linear Equations

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Types of functions

• vi. Polynomial Functions
• vii. The Quadratic Function
• viii. The Cubic Function
• ix. The roots of a quadratic function x1, 2
  -b ? ?(b² - 4ac) / 2a
• x. Power functions

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Types of functions

• xi. Transcendental functions - Logarithmic
  functions
• xii. Functions of several variables

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Calculus

• Calculus primarily involves two kinds of
  operations (which are complementary)
  differentiation and integration.
• Differentiation is a way to calculate
  instantaneous change (e.g. the change in the
  speed of a car).
• Integration, on the other hand, is a way to
  calculate the distance the car has passed given
  its speed at different points in time.
• In order to understand those operations we should
  be familiar with the concepts of limits and
  continuity.

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Limits and continuity

• The concept of a limit
• A formal definition - lim(x?a) f(x)A iff for
  any chosen positive number, ?, however small,
  there exists a positive number ? such that
  whenever 0 lt ?x-a? lt ?, then ?f(x) - A?lt?.
  (Ayers and Mendelson, 2000)
• So, in plain English --- in order to have a limit
  we should have for every number in the small
  neighbourhood of a (the chosen value of x) a
  corresponding number in the small neighbourhood
  of A (the chosen value of y)

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Theorems on limits

• a constant functionf(x) c, then lim(x?a)f(x)
  c
• If lim(x?a)f(x) A and lim(x?b)f(x) B then
• lim(x?a)k?f(x) kA (k is a constant of some
  form)
• lim(x?a)f(x) ? g(x) lim(x?a)f(x) ?
  lim(x?a)g(x) A ? B
• lim(x?a)f(x) ? g(x) lim(x?a)f(x) ?
  lim(x?a)g(x) A ? B
• lim(x?a)f(x) / g(x) lim(x?a)f(x) /
  lim(x?a)g(x) A / B (B?0)
• lim(x?a) n? f(x) n? lim (x?a)f(x) n?A (n?A
  should be a real number)
• lim(x?a) xn an, n is a positive integer
• lim(x?a) P(x) P(a), where P is any polynomial
  function

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Limits at infinity and infinite limits

• Infinite limits of functions
• Limits of functions at infinity

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Asymptotes

• 1. Vertical Asymptotes
• Vertical asymptotes correspond to the x values
  for which the denominator of a rational function
  equals zero.
• More generally, we find vertical asymptotes where
  - lim (x?a)f(x) ??
• 2. Horizontal Asymptotes
• Horizontal asymptotes usually indicate behavior
  at x??. Thus, they indicate general behavior far
  off to the sides of the graph.
• More generally, we find vertical asymptotes where
  - lim (x??)f(x) A (the value of the asymptote)
• Thus

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Continuity

• A function is continuous iff it is continuous at
  every point of its domain.
• There are three conditions for the continuity of
  the function at a point (say point a)
• f(x) is defined at x a
• lim (x?a) f(x) must exist
• lim (x?a) f(x) f(a)

Continuity
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Rules for combining continuity

• If f(x) and g(x) are continuous at a, then
• f g and f g are continuous at a
• f ? g is continuous
• provided that g(a) ? 0, f / g is continuous
• f(x)r/q is continuous at a if f(x)r/q is
  defined
• If f is continuous at g(c) and g is continuous at
  c, then so is f(g(c))

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Theorems of well-behaved functions (Part I)

• Extreme value theorem
• Browers fixed point theorem
• Optimum point theorem (Implied by Intermediate
  Value Theorem)
• Intermediate value theorem

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Differential calculus

• derivative function

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Rules for differentiating

• i. Constant functions
• f(x) c ? f ?(x) 0
• ii. Linear functions
• f(x) mx b ? f ?(x) m --- i.e. the slope of
  the function is its derivative
• iii. Derivative of a constant times a function
• In such a case the derivatives equals the
  constant times the derivative of the function.
• c f(x)? c f(x)?

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Rules for differentiating

• iv. Monomial functions the Power Rule
• f(x) xn f(x)? n xn-1
• v. Sum of two functions
• (f(x) g(x))? f ?(x) g?(x)
• vi. Product rule
• (f(x) ? g(x))? f ?(x) ? g(x) f(x) ? g?(x)
• vii. Quotient rule
• (f(x) / g(x))? f ?(x) ? g(x) - f(x) ? g?(x) /
  (g(x))²

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Rules for differentiating

• viii. The chain rule composition of two
  functions
• (f(g(x)))? f ?(g(x)) ? g?(x)
• ix. Exponents
• f(x) ex ? f ?(x) ex
• g(x) ax ? g?(x) ln a ? ax
• x. Logarithms
• f(x) ln x ? f ?(x) 1 / x
• g(x) log a x ? f ?(x) 1 / (x ln a)

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• Higher order derivatives
• Multivariate functions and partial derivatives
• the partial derivative for x would be?f(x, y) /
  ?x lim (h?0) (f(xh, y) f(x, y)) / hAnd
  for y?f(x, y) / ?y lim (h?0) (f(x, yh)
  f(x, y)) / h

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Theorems of well-behaved functions (Part II)

• The Extreme Value Theorem
• Fermat's Theorem
• Rolles Theorem
• Mean Value Theoremf '(c) (f(b) f(a)) / (b-a)