least squares approximation

Examples of least squares approximation in the following topics:

• Fitting a Curve

  • The simplest and perhaps most common linear regression model is the ordinary least squares approximation.
  • This approximation attempts to minimize the sums of the squared distance between the line and every point.  
  • Example:  Write the least squares fit line and then graph the line that best fits the data 
  • The line found by the least squares approximation, $y = 0.554x+0.3025$.
  • Model a set of data points as a line using the least squares approximation

• Radical Functions

  • If the square root of a number is taken, the result is a number which when squared gives the first number.
  • Roots do not have to be square.
  • However, using a calculator can approximate the square root of a non-square number:$\sqrt {3} = 1.73205080757$
  • The result of taking the square root is written with the approximately equal sign because the result is an irrational value which cannot be written in decimal notation exactly.
  • Writing the square root of 3 or any other non-square number as $\sqrt {3}$ is the simplest way to represent the exact value.

• Domains of Rational and Radical Functions

  • To solve, divide both sides by $2$, add $5$ to both sides, and then take the square root of both sides to yield:
  • Therefore the domain is the set of all real numbers except the square root of five or negative square root of five.
  • At the values of $x=\pm \sqrt { 5 }$ (which is approximately $\pm 2.2$), the graph does not exist.
  • The principal square root function $f(x)=\sqrt x$ (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself.  
  • Note that half of the parabola is missing since functions cannot have more than one value at a point, and the square root function is taken to yield a positive value (though $(-x)^2$ gives the same value as $x^2$ so the square root of a number $y$ such that $y=x^2$ would be $\sqrt y = \pm x$).

• Solving General Problems with Logarithms and Exponents

  • . √9 asks the question "What number squared is 9?
  • " So the equation √9 = 3 asks this question, and then answers it: "3 squared is 9. "
  • It will take approximately 45 years for the population to triple in size.

• Applications of the Parabola

  • What Galileo discovered and tested was that when gravity is the only force acting on an object, the distance it falls is directly proportional to the time squared.
  • As in all cases in the physical world, a projectile's trajectory is an approximation.
  • The presence of air resistance, for instance, distorts parabolic shape, although at low speeds the shape is a good approximation.
  • This is because air resistance is a force acting on the object, and is proportional to the object's area, density, and speed squared.

• The Number e

  • The number $e$ is an important mathematical constant, approximately equal to $2.71828$.
  • The number $e$, sometimes called the natural number, or Euler's number, is an important mathematical constant approximately equal to 2.71828.
  • Another is that $e$ is the unique number so that the area under the curve $y=1/x$ from $x=1$ to $x=e$ is $1$ square unit.

• Circles as Conic Sections

  • The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654.
  • As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference, and whose height equals the circle's radius, which comes to π multiplied by the radius squared:
  • that is, approximately 79 percent of the circumscribing square (whose side is of length d).