Math Warm Up http://math.expectable.net Simple calculus problems Sun, 05 Apr 2009 08:11:38 +0000 http://wordpress.org/?v=2.7.1 en hourly 1 Derivatives http://math.expectable.net/derivatives http://math.expectable.net/derivatives#comments Fri, 06 Mar 2009 22:02:35 +0000 Jack Leeman http://math.expectable.net/?p=176 A derivative is a “rate of change” of a continuous function at a point in a coordinate system. The derivative of function f at point a is defined by:

.    (1)

For the purpose of explaining the rules of derivatives below, we also use to represent .

Rules of Derivatives
  1. where c = constant => .   (2)
  2. .   (3)
  3. .   (4)
  4. .   (5)
  5. .   (6)
  6. The Chain Rule

        Consider 2 differentiable functions:

        and .

    =>  .   (7)

Geometric interpretation of a 1st derivative (e.g. ) is the slope of the tangent line. A 2nd derivative () has the value of 0 at the point of inflection.

Derivative Graph 1
Fig. 1


Continue to Part 2
]]> http://math.expectable.net/derivatives/feed Exponential/Logarithmic Functions http://math.expectable.net/exponentiallogarithmic-functions http://math.expectable.net/exponentiallogarithmic-functions#comments Fri, 06 Mar 2009 22:22:13 +0000 Jack Leeman http://math.expectable.net/?p=179 The Exponential function and Logarithmic function, with the base of are related like this:

.   (1)

Laws of Exponents
  1. .   (2)
  2. .   (3)
  3. .   (4)

Using the fact that

  • ,   (5)
we can easily get
  • .   (6)
.   (7)


Laws of Logarithms

  1. .   (8)
  2. .   (9)

It can be easily derived that
  •    (10)

If we replace the base with e ≈ 2.71828…, then logarithm is commonly written as


The Laws of Logarithms can be rewritten:
  •    (11)
  •    (12)
  • .   (13)

Continue to Part 2
]]> http://math.expectable.net/exponentiallogarithmic-functions/feed Vectors http://math.expectable.net/vectors http://math.expectable.net/vectors#comments Fri, 06 Mar 2009 22:33:59 +0000 Jack Leeman http://math.expectable.net/?p=182 A vector, as opposed to a scalar, has a direction.. For example, velocity is a vector while speed is a scalar. When one says a car drives “at 60 mi/hr,” he/she is describing its “speed.” On the other hand, “60 mi/hr eastbound” is “velocity.”

Vector Graph 1





Vector Graph 2




or


The addition can be easily understood by breaking down each vector into components which are “projections” of the vector onto the x-y coordinates.

we can see that
and the direction of C points to .

The length of the vector is .


Product of Vectors
  1. Dot Product
    • Let A and B be 3-dimensional vectors represented in a rectangular coordinate system as:
      • ,
      • .
    • When i, j, k are unit vectors in the x, y, z directions respectively.


    • The dot product is defined as
      • .

    • Properties of the dot product
      • , where c is a scalar.

    • It can be proven that
      • , where θ is the angle betweeen A and B.
      • and
      • are the lengths of the respective vectors.


      It is obvious that the dot product of 2 orthogonal vectors equals 0.

  2. Cross Product

    • The cross product can be written as:

    • The value of the 3rd-order determinant is calculated:
      • ,

    • where the 2nd-order determinants are calculated:

    • It can be proven that
      • , where θ is the angle between the z vectors A and B.


    • The direction of the cross product can be memorized using the “rule of right thumb”:

    • It is obvious from the rule of right thumb that A × B points to the opposite direction of B × A. Since = 0 when θ = 0, cross product of 2 parallel vectors is 0.

    • Properties of Cross Products:
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