Math Camp 2005 Instructor: Udi Sommer

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Math Camp 2005Instructor Udi Sommer

• Fifth Class (Thursday, August 25, 10a-1p)
• Integration
• Indefinite integrals
• Definite integrals
• Formula
• Method of substitution
• Integration by parts
• Areas by integration
• Matrix algebra

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Integrals

• Integration, or antidifferentiation, is the
  inverse of differentiation
• In Biondis example, we had information about his
  distance as a function of time. The derivative
  function of this function provided us with
  information about the instantaneous change in the
  distance which is his velocity.
• Lets say we had a different kind of information
  we had information about his speed at every
  point in time --- and --- we were interested in
  the distance he swam.
• The mathematical tool that we could use here is
  integration.
• In terms of the curve or the graph, we can think
  of it as followsWe are provided with the graph
  describing the function of Biondis speed.When
  we integrate we get the area under that curve,
  which is the distance.

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Indefinite integral

• Definition a function F is the indefinite
  integral of a function on an interval I iff
  F?(x) (x) for all x?I (Hagle, 1995)
• So, if we take the derivative of F we should get
  .
• Indefinite integrals are not unique.

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Notation

• ? (x)dx F(x) c
• ? is the integral sign
• (x) is the integrand
• c is the constant of integration
• dx indicates that the integration is taking place
  with respect to a particular variable, x in this
  case. In other words, this is the differential.

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Integration rules

• Power rule for integration?xndx (xn1 /
  (n1)) c
• Integral of x-1? x-1dx ln? x? c
• Integral of 1, or (by extension) of any other
  constant? dx x c

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Integration rules

• IV. A constant times a function? a ? (x)dx a
  ? ? (x)dx
• V. Sum of (or difference between) two functions?
  (x) ? g(x) dx ? (x)dx ? ? g(x)dx
• VI. The chain rule for integration? (g(x)) ?
  g?(x) dx Fg(x) c

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Integration rules

• VII. Integration by parts? g(x) ? f(x) dx
  g(x) ? F(x) - ? g?(x) ? F(x) dx c
• VIII. Exponents and logarithms rules? ex dx ex
  c ? ax dx (ax / ln a) c

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Definite Integrals

• We use definite integrals to determine the area
  under a curve on a specific interval.
• For that purpose we use the fundamental theorem
  of calculus a?b (x) dx F(x)?ab f(b) F(a)
• This formula means that the area under a curve
  equals the difference between the value of the
  integral at one endpoint and the value of the
  integral at the other endpoint.

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Important rules and results (Hagel, 1995 Ayers
Mendelson, 2000)

• if a function is continuous on interval a, b it
  is integrable on a, b
• a?b (x) dx - b?a (x) dx
• a?a (x) dx 0
• a?b dx b-a

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Important rules and results

• 5. a?b (x) dx ? g(x) dx a?b (x) dx ? a?b
  g(x) dx
• 6. for a lt c lt b it is true thata?b (x) dx
  a?c (x)dx c?b (x) dx
• 7. The average value of function on a, b is
  (a?b (x) dx) / (b a)

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Matrix algebra

• A matrix is an ordered rectangular array of
  numbers or elements
• The element aij is the value of the entry in the
  ith row and the jth column, counting from the
  upper left corner.
• Order the order of the matrix is determined by
  its dimensions. Am?n An m by n matrix is a
  matrix with m rows and n columns.
• A square matrix An?n this matrix has the same
  number of rows and columns.
• A column vector - Am?1 this matrix has only one
  column
• A scalar A1?1 is simply a number
• Transposition the transpose of Am?n, denoted
  A?, is an n?m matrix obtained by interchanging
  the rows and the columns of A (Gujarati, 1995)-
  The transpose of a row vector is a column vector
  and vice versa.- Transpose of a transpose is the
  matrix itself
• Submatrix if all but r rows and s columns of A
  are deleted, the resulting matrix of order r?s is
  called a submatrix of A.

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Types of matrices

• A square matrix An?n this matrix has the same
  number of rows and columns.
• Diagonal matrix a square matrix with at least
  one nonzero element on the main diagonal and
  zeroes elsewhere is called a diagonal matrix.
• Scalar matrix this is a diagonal matrix whose
  diagonal elements are all equal.
• Identity matrix OR Unit matrix this is a
  diagonal matrix whose diagonal elements are all
  1.
• Symmetric matrix this is a square matrix whose
  elements above the main diagonal are mirror
  images of the elements below the main diagonal.
• Null matrix a matrix whose elements are all
  zero. It is denoted by 0.
• Null vector a row or a column vector whose
  elements are all zero. It is also denoted by 0.

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Operations

• Equality -Two matrices A aij and B bij
  are said to be equal if they have the same order
  and aij bij for all i and j.
• Addition For A aij and B bij with equal
  order, m ? n, the matrix AB is of the order m ?
  n such that A B aij bij
• Multiplication by a scalar -If Aaij and k is
  a scalar then kA kaij
• Matrix subtraction is thusA B A (-1)B
• Matrix multiplication For matrices to be
  conformable with respect to multiplication, the
  of columns in A must be equal to the of rows in
  B.- a row vector postmultiplied by a column
  vector is a scalar- a column vector
  postmultiplied by a row vector is a matrix-
  matrix multiplication is distributive with
  respect to addition - A(B C) AB AC

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Operations

• Matrix inversion -An inverse of a square matrix
  A, denoted by A-1 (A inverse), if it exists, is a
  unique square matrix such that AA-1 A-1A
  I
• Determinants -To every square matrix, A, there
  corresponds a number (scalar) known as the
  determinant of the matrix detA or ?A?.-
  Evaluation of a 2?2 determinant - ?A? a11?a22
  a12?a21- evaluation of a 3?3 determinant -?A?
  a11a22a33 a11a23a32 a12a23a31 a12a21a33
  a13a21a32 a13a22a31
• Rank of a matrix -The rank of a matrix is the
  order of the largest square submatrix whose
  determinant is not zero
• Minor -If we delete the ith row and the jth
  column of a square matrix, the determinant of the
  resulting matrix is called the minor of the
  element aij, which is the element in the
  intersection of the ith row and the jth column.