objective function

Examples of objective function in the following topics:

• Application of Systems of Inequalities: Linear Programming

  • The function to be maximized (the objective function, and in this case, the profit on the chairs) is:
  • Standard form also requires the objective function to be a minimization.
  • If the problem calls for maximization, multiply the objective function by $-1$.
  • where the first row defines the objective function and the remaining rows specify the constraints.
  • Therefore, the objective function is unbounded.

• Symmetry of Functions

  • Two objects have symmetry if one object can be obtained from the other by a transformation.
  • Two objects are symmetric to each other with respect to the invariant transformations if one object is obtained from the other by one of the transformations.
  • In the case of symmetric functions, determining symmetry is as easy as graphing the function or evaluating the function algebraically.  
  • Symmetry of a function can be a simple shift of the graph (transformation) or the function can be symmetric about a point, line or axes.
  • Functions and relations can be symmetric about a point, a line, or an axis.  

• 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.
  • For objects extended in space, such as a diver jumping from a diving board, the object follows a complex motion as it rotates, while its center of mass forms a parabola.
  • This is because air resistance is a force acting on the object, and is proportional to the object's area, density, and speed squared.
  • For dense objects, and/or at low speeds, the air resistance force is small.
  • All of the physical examples are situations where an object's trajectory or the shape of an object fits a generalized parabola function:

• Recursive Definitions

  • A recursive definition of a function defines its values for some inputs in terms of the values of the same function for other inputs.
  • In mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself.
  • A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs.
  • For example, the factorial function $n!

• Scientific Applications of Quadratic Functions

  • If any of the variables $a$, $b$, or $c$ represent functions in themselves, it is often useful to expand the terms, combine like variables, and then re-factor the expression.
  • Consider the equation relating gravitational force ($F$) between two objects to the masses of each object ($m_1$ and $m_2$) and the distance between them ($r$):
  • The shape of this function is not a parabola, but becomes such a shape if rearranged to solve for $m_1$ or $m_2$, as seen below:
  • Substituting any time ($t$) in place of $t_h$ leaves the equation for height as a quadratic function of time.

• Functions and Their Notation

  • We write the function as:$f(-3)=9$.
  • In the case of a function with just one input variable, the input and output of the function can be expressed as an ordered pair.
  • Let's say the machine has a blade that slices whatever you put in into two and sends one half of that object out the other end.
  • All functions are relations, but not all relations are functions.
  • Connect the notation of functions to the notation of equations and understand the criteria for a valid function

• Transformations of Functions

  • A reflection of a function causes the graph to appear as a mirror image of the original function.  
  • Let the function in question be $f(x) = x^5$.  
  • A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation.  
  • Where $x_1$and $y_1$are the new expressions for the rotated function, $x_0$ and $y_0$ are the original expressions from the function being transformed, and $\theta$ is the angle at which the function is to be rotated.  
  • If we rotate this function by 90 degrees, the new function reads:

• Models Involving Nonlinear Systems of Equations

  • Its position in meters (y) can be determined as a function of time in seconds (t), by the formula:
  • Its position (y) in meters can be determined as a function of time (t) in seconds, using the following formula:
  • To express the position of the second car relative to the first as a function of time, we can modify the second equation as such:
  • The kinetic energy of the objects depends on the speed squared, and the momentum depends on the speed directly.
  • Rendering and visualizing these objects, and formulating a plan for constructing them, requires the software to solve nonlinear systems.

• Introduction to Sequences

  • A mathematical sequence is an ordered list of objects, often numbers.
  • In mathematics, a sequence is an ordered list of objects.
  • A more formal definition of a finite sequence with terms in a set $S$ is a function from $\left \{ 1, 2, \cdots, n \right \}$ to $S$ for some $n > 0$.
  • An infinite sequence in $S$ is a function from $\left \{ 1, 2, \cdots \right \}$ to $S$.
  • For example, the sequence of prime numbers $(2,3,5,7,11, \cdots )$ is the function $1→2, 2→3, 3→5, 4→7, 5→11, \cdots$ .

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • A rational function is any function which can be written as the ratio of two polynomial functions.
  • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.