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• Operations on functions
  '''Example (intercepts).''' For a function $F:{\bf R}\to {\bf R}$, its graph is the following set given presented via Suppose $y=F(x)$ is a numerical function. Then the $x$-''intercepts'' of $F$ are the elements of the intersection of
  142 KB (23,566 words) - 02:01, 23 February 2019
• Functions
  [[image:boys and balls -- relation and function.png| center]] [[image:boys and balls -- function.png| center]]
  151 KB (25,679 words) - 17:09, 20 February 2019
• Precalculus
  *1. finding the distance between two points, and For example, suppose $P$ is a ''location'' on the line. We then find the distance from the origin -- positive in the positive direction and negative in the n
  100 KB (16,148 words) - 20:04, 18 January 2017
• Calculus I, the discrete version
  *a node function $f: 0\mapsto 2,\ 1\mapsto 4,\ 2\mapsto 3,\ ...$; and *an edge function $s: [0,1]\mapsto 3,\ [1,2]\mapsto .5,\ [2,3]\mapsto 1,\ ...$.
  64 KB (11,521 words) - 19:48, 22 June 2017
• Classes of functions
  Now, what if ''all'' boys prefer basketball? Then the “preference function” $F$ cannot be simpler: all of its values are equal and all arrows point The table of this function $F$ is also very simple: all crosses are in the same column; and the graph
  143 KB (24,052 words) - 13:11, 23 February 2019
• Differentiation
  ...What this means is that this procedure is a special kind of function, a ''function of functions'': ...hat this means is that this process is a special kind of function too, a ''function of functions'':
  82 KB (14,116 words) - 19:50, 6 December 2018
• Functions of several variables
  ...ons. On the other hand, we can see that the surface that is the graph of a function of two variables produces -- through cutting by vertical planes -- ''infini We represent a function diagrammatically as a ''black box'' that processes the input and produces t
  97 KB (17,654 words) - 13:59, 24 November 2018
• Limits and continuity
  One of the most crucial properties of a function is the integrity of its graph: ''is there a break or a cut?'' For example, If there is a jump in the graph of the function, it can't represent motion!
  107 KB (18,743 words) - 17:00, 10 February 2019
• Functions in higher dimensions
  ...early calculus (Chapters 7 -13) we deal with only numbers, the graph of a function of one variable lies in the $xy$-plane, a space of dimension $2$. [[image:function of two variables -- heat map.png| center]]
  113 KB (19,680 words) - 00:08, 23 February 2019
• Applications of integral calculus
  ...preadsheet, $\sum_i f(c_i)\cdot.1$, and them subtract the data for the new function, $\sum_i g(c_i)\cdot.1$. Furthermore, we have ...he following. We ''recognize'' this expression as the Riemann sum of a new function, $f-g$:
  103 KB (18,460 words) - 01:01, 13 February 2019
• Calculus exercises: part I
  It's just a limit. But we recognize that this is the derivative of some function. We compare the expression to the formula in the definition: The function is computed in two steps. Indeed, if
  49 KB (8,436 words) - 17:14, 8 March 2018
• Differential calculus of parametric curves
  A parametric curve is such a function: ...the latter vector, $OX$. In either case, this is just a combination of two function of the same independent variable.
  130 KB (22,842 words) - 13:52, 24 November 2018
• The derivative
  We approached the problem by plotting the location as a function of time: [[image:location as a function of time.png| center]]
  75 KB (13,000 words) - 15:12, 7 December 2018
• Differential calculus
  *maximize the function $A(W)=-W^2+50W$. [[image:cattle -- function 2.png| center]]
  84 KB (14,321 words) - 00:49, 7 December 2018
• Several variables
  ...formulas can now be solved in order to be able to model the location as a function of time. The result is these recursive formulas for the ''Riemann sums'': ...00$ and $0$ respectively. Below, the velocity is computed as a Riemann sum function of the previous column, with the same formula:
  76 KB (13,017 words) - 20:26, 23 February 2019
• Vector and complex variables
  ...real-valued functions of two variables. Consider $u=f(x,y)=2x-3y$, such a function: Consider another such function: $v=g(x,y)=x+5y$ is also a real-valued function of two variables:
  113 KB (18,750 words) - 02:33, 10 December 2018
• The integral
  ...nfirm the formula with nothing but a spreadsheet. We plot the graph of the function: ...e development of algebra, the Cartesian coordinate system, and the idea of function (Chapters 2, 3, and 4).
  66 KB (11,473 words) - 21:36, 19 January 2019
• Precalculus exercises
  ...)=x^2+3x-10$. Find the $x$- and $y$-intercepts and sketch the graph of the function. *What is the distance from the center of the circle $(x-1)^2+(y+3)^2=5$ to the origin?
  17 KB (2,933 words) - 19:37, 30 July 2018
• Freshman's introduction to discrete calculus
  *a node function $f: 0\mapsto 2,\ 1\mapsto 4,\ 2\mapsto 3, ...$; and *an edge function $s: [0,1]\mapsto 3,\ [1,2]\mapsto .5,\ [2,3]\mapsto 1, ...$.
  42 KB (7,443 words) - 14:18, 1 August 2016
• Vector fields
  *$r$ is the distance between the centers of the masses. That's the vector form of the law! We plot the magnitude of the force as a function of two variables:
  91 KB (16,253 words) - 04:52, 9 January 2019
• Integration
  Suppose a function $f$ is defined on an open interval $I$. Then a function $F$ defined on $I$ that satisfies $F' = f(x)$ for all $x$ is called an ''an ...eorem of Calculus).''' (I) Given a continuous function $f$ on $[a,b]$, the function defined by
  69 KB (11,727 words) - 03:34, 30 January 2019
• Calculus of chain maps
  First, $f$ has to be a function that takes nodes to nodes: ...h first and then attach edges to them. Therefore, we require from the edge function $f$ the following:
  41 KB (7,344 words) - 12:52, 25 July 2016
• Series
  ...t is called its best linear approximation and its happens to be the linear function the graph of which is the tangent line at the point. The replacement is jus However, there is a more basic approximation: a constant function, $y=C(x)$.
  113 KB (19,100 words) - 23:07, 3 January 2019
• Continuous functions
  First we, informally, discussed continuity of a function as a transformation that does not tear things apart and interpreted this id <!--200-->[[Image:continuous function.png|center]]
  42 KB (7,138 words) - 19:08, 28 November 2015
• Discrete calculus article
  ...ving the squaring function turns out to be something close to the doubling function. ...xteen, and so on—and uses this information to output another function, the function $g(x)=2x+h$, as will turn out. It is defined at the middle points of the ab
  27 KB (4,329 words) - 16:02, 1 September 2019
• Applications of differential calculus
  Given a function $y=f(x)$, find such a $d$ that $f(d)=0$. We have a function $f$ is defined and is continuous on interval $[a,b]$ with $f(a)<0,\ f(b)>0$
  59 KB (10,063 words) - 04:59, 21 February 2019
• Differential equations
  ...re, the ''difference'' of a function $y$ defined at the primary nodes is a function defined at the secondary nodes of the partition: We can also think of this sequence as a function defined at the nodes of the partition:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Functions of several variables: derivatives and integrals
  ==The derivative of a function of several variables== The linear approximations of a function $z=f(X)$ at $X=A$ in ${\bf R}^n$ are linear functions with $n$ slopes in th
  42 KB (6,904 words) - 15:15, 30 October 2017
• Applications of ODEs
  *the unit vector in this direction (dividing by the distance between them); ...the mass is equal to $1$. Then the ''kinetic energy'' is known to be this function of time:
  50 KB (8,692 words) - 14:29, 24 November 2018
• Partial differential equations
  ...id this “overshoot”, the increment of the heat shouldn't be more that half-distance to the heat of the other room. This is the result of our simulation is the collection of graphs of the function $u(t,\cdot)$ of one variable for each $t$ (the graph of $u$ is of course a
  53 KB (9,682 words) - 23:19, 18 November 2018
• Discrete forms and cochains
  ...$ is often thought of as a function the input of which is any integrable ''function'' $f$ while the output is a real number. This idea is revealed by the usual ...the limit of the Riemann sums of $f$. The student then discovers that this function is ''linear'':
  34 KB (5,619 words) - 16:00, 30 November 2015
• Topological spaces
  There are no measurements in topology. Does the distance between a point and a set make any sense? Let's just try to decide if the distance is $0$ or not.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Integrals of functions of several variables
  *the height of the bar in this rectangle equal to the value of the function and with the ones outside the domain replaced with $0$s, and Suppose a function $y = f(X)=f(x,y)$ defined at the tertiary nodes of the partition of the rec
  73 KB (13,324 words) - 14:06, 24 November 2018
• Calculus of sequences
  ...{ speed }= \text{ distance } / \text{ time }\quad \text{ and }\quad \text{ distance }=\text{ speed }\times \text{ time }. \quad \\ \hline\end{array}$$ The formula is solved for the distance or for the speed depending on that is known and what is unknown.
  113 KB (18,425 words) - 13:42, 8 February 2019
• Discrete forms
  '''Definition.''' A ''cubical'' $k$-''form'' is a function defined on $k$-cells. To emphasize the nature of a form as a function, we can use arrows:
  35 KB (6,055 words) - 13:23, 24 August 2015
• Relative topology
  ...distance formula, the Euclidean metric<!--\index{Euclidean metric}-->. The distance between $(x,y)$ and $(a,b)$ is '''Theorem.''' Suppose $f : X \to Y$ is continuous<!--\index{continuous function}-->. If $X$ is path-connected<!--\index{path-connectedness}--> the so is $f
  34 KB (6,089 words) - 03:50, 25 November 2015
• Calculus on cubical complexes
  ...''' A ''cubical''<!--\index{cubical form}--> $k$-''form'' is a real-valued function defined on $k$-cells of ${\mathbb R}^n$. To emphasize the nature of a form as a function, we can use arrows:
  36 KB (6,218 words) - 16:26, 30 November 2015
• Applications of derivative: optimization
  Substitute to create a function of a ''single'' variable: Eliminate the extra variables to create a function of single variable to be maximized or minimized.
  6 KB (891 words) - 02:15, 17 July 2011
• Discrete parametric curves
  Consider the distance formula in ${\bf R}^2$. Then, the distance from $d=(a,b)$ to $0$ is $\sqrt{a^2+b^2}$.
  32 KB (5,426 words) - 21:57, 5 August 2016
• Topology Illustrated -- Index
  *[[constant function|constant function]] *[[continuous function|continuous function]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Introduction to discrete calculus
  ...es of $1$-chains are $0$-chains as well. Then, boundaries are found by the function defined above: ...e think initially of a <!--\index{cochain}--> $k$-cochain as a real-valued function defined on $k$-cells of ${\mathbb R}$.
  40 KB (6,983 words) - 19:24, 23 July 2016
• Vector integrals
  Given motion with the position function $F : {\bf R} {\rightarrow} {\bf R}^n$ during a time interval $[ a, b ]$, we [[Image:lengths - displacement and distance traveled.jpg|right]]
  15 KB (2,545 words) - 19:47, 20 August 2011
• Treat calculus discretely!
  Let's plot our location as a function of time: [[image:location as a function of time.png| center]]
  8 KB (1,196 words) - 12:02, 4 July 2018
• Homeomorphism
  Homeomorphisms are [[continuous functions]]<!--\index{ continuous function}--> that preserve topological properties. That's why we will define a class of maps so that both the function
  13 KB (2,168 words) - 13:09, 7 August 2014
• Metric tensor
  ..., or deforming the $x$- or the $y$ axes won't change the monotonicity of a function but it will change its concavity. '''Proposition.''' Given a differentiable function $f:{\bf R}\to {\bf R}$,
  42 KB (7,131 words) - 17:31, 30 November 2015
• Maps of polyhedra
  ...f such as approximation is a ''sequence'' of polynomials converging to the function. This time the goal is ...nts will be allowed. Instead, we start with the simple idea of using “cell distance” as one and only “measurement” of closedness.
  51 KB (9,162 words) - 15:33, 1 December 2015
• Inner product spaces: part 1
  Consider the [[distance formula]] in ${\bf R}^2$. Then, the distance from $d=(a,b)$ to $0$ is $\sqrt{a^2+b^2}$.
  14 KB (2,404 words) - 15:04, 13 October 2011
• Binary watershed
  ...geJ.jpg|(1) the original image of particles that touch each other, (2) the distance of each pixels to the nearest white pixel is illustrated with its gray leve ...est white pixel. This is called the [[distance function]]. It’s a [[scalar function]] of two variables.</p>
  5 KB (747 words) - 22:05, 16 May 2010
• Sequences and their limits
  Even though every function $y=f(x)$ with an appropriate domain creates a sequence, $a_n=f(n)$, the con A function defined on a ray in the set of integers, $\{p,p+1,...\}$, is called an ''in
  64 KB (10,809 words) - 02:11, 23 February 2019
• Convergence of the discrete to the continuous
  The diagram commutes. Indeed, given a function $f:{\bf R}\to {\bf R}$, we can proceed in two ways: *right then down: we acquire a $0$-form $g$ by sampling function $f$, and then we acquire $dg$ by taking the differences of the values of $g
  21 KB (3,664 words) - 02:02, 18 July 2018

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