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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
• 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
• 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
• 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
• 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
• Precalculus
  ...the ''signs'' of these numbers, it can be restated in terms of the ''sign function'': Such an implicit relation between two variables is called a ''function''. This is the data:
  100 KB (16,148 words) - 20:04, 18 January 2017
• 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
• 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
• 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
• 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
• 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
• Precalculus exercises
  ...)=x^2+3x-10$. Find the $x$- and $y$-intercepts and sketch the graph of the function. ...$55$ and its leading term is $-1$. Describe the long term behavior of this function.
  17 KB (2,933 words) - 19:37, 30 July 2018
• Preview of calculus: part 3
  == Exponential Function == ==Exponent as a function==
  17 KB (2,498 words) - 15:06, 19 March 2011
• 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
• 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
• 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
• Limits: transition from discrete to continuous
  A function defined on a ray in the set of integers, $\{p,p+1,...\}$, is called an (inf Algebraically, we see that for every measure of closeness $\varepsilon$, the function's values become eventually that close to the limit.
  51 KB (9,271 words) - 20:02, 8 September 2016
• Systems of ODEs
  Every solution $x=x(t)$ and $y=y(t)$, when substituted into the function We turn instead to the actual function. First, we plot it with a spreadsheet ($\alpha=3$, $\beta=2$, $\gamma=3$, $
  63 KB (10,958 words) - 14:27, 24 November 2018
• College Algebra -- Fall 2016
  '''MTH 130 College Algebra.''' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Graphs, systems of equations and inequalities, * 7.1 Exponential Functions and their Graphs
  10 KB (1,078 words) - 19:07, 16 December 2016
• ODEs
  where $G:R\to R$ is some function and $q:C_1({\mathbb R})\to C_0({\mathbb R})$ is given by The growth is exponential (geometric), as expected.
  47 KB (8,415 words) - 15:46, 1 December 2015
• Precalculus with Scientific Applications -- Spring 2016
  ...plications.''' Functions used in calculus including polynomial, rational, exponential, logarithmic, and trigonometric. Systems of equations and inequalities, con ...ting Information from the Graph of a Function. Average Rate of Change of a Function. Transformations of Functions. Combining Functions. One-to-One Functions an
  7 KB (890 words) - 16:32, 20 April 2016
• Calculus Illustrated
  10 A function as a black box 11 Give the function a domain...
  16 KB (1,933 words) - 19:50, 28 June 2021
• College Algebra -- Fall 2013
  '''MTH 130 College Algebra.''' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Graphs, systems of equations and inequalities, 3.2 The Graph of a Function
  6 KB (752 words) - 04:19, 13 December 2013
• College Algebra -- Fall 2014
  ''MTH 130 College Algebra.'' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Graphs, systems of equations and inequalities, 3.2 The Graph of a Function
  6 KB (850 words) - 16:52, 29 November 2014
• Implicit differentiation
  $$ \underbrace{x^{2}}_{\text{function}} = \underbrace{1}_{\text{Number}} \to \text{ find a particular number} x$$ $$ u^{2} = \sin x \to \text{ find a particular function } u$$
  9 KB (1,445 words) - 15:50, 2 May 2011
• Calculus by Rogawski
  1.5 [[Inverse function|Inverse Function]]s 1.6 [[Exponential and logarithmic functions|Exponential and Logarithmic Functions]]
  6 KB (634 words) - 16:38, 1 March 2013
• Continuity of functions
  Now graphically, what is the mathematical idea of continuity? A [[continuous function]] should have a "continuous" graph, i.e. one made up of one piece. What doe Let's consider this in detail. Given a point $a$ and a function $f$, we have three non-overlapping pieces:
  10 KB (1,839 words) - 00:26, 25 September 2013
• College Algebra by Sullivan
  3.2 The Graph of a Function 5.3 The Graph of a Rational Function
  3 KB (349 words) - 16:29, 8 August 2013
• College Algebra by Ratti and McWaters
  3.4 The Real [[Zeros]] of a Polynomial Function 3.5 The Complex Zeros of a Polynomial Function
  2 KB (269 words) - 18:53, 16 November 2011
• Applications of ODEs
  As $\lambda_{1},\lambda_2<0$, we have exponential decay (a stable node of the vector ODE): the friction brings the ODE back t Again, as $\lambda=\lambda_{1}=\lambda_2<0$, we see exponential decay (a stable improper node of the ODE): the ODE's motion dies out, even
  50 KB (8,692 words) - 14:29, 24 November 2018
• Homotopy
  ...d” into any other. In fact, a simpler idea is to push the graph of a given function $f$ to the $x$-axis: <center>homotopy is a continuous transformation of a continuous function.</center>
  46 KB (7,846 words) - 02:47, 30 November 2015
• Calculus by Stewart
  .... Graphing calculators and computers. [[Exponential function]]s. [[Inverse function]]s and [[logarithm]]s. ...Writing project: early methods for finding tangents. The [[derivative as a function]].
  6 KB (794 words) - 16:29, 13 August 2017
• Exponential models
  ; Growth : Know if $k > 0$, then this function is increasing and $\lim\limits_{x \to \infty} y(x) = \infty $. ; Decay : if $k < 0$, the this function is decreasing and $\lim\limits_{x \to \infty} y(x) = 0$.
  8 KB (1,201 words) - 15:45, 2 May 2011
• Fantasy math
  Sometimes there is a [[continuous function]] or process behind the numbers but often there isn't. The issues one has t What is commonly done is to go back to [[continuous function]]s via [[approximation]], [[interpolation]], [[curve fitting]], etc. This a
  8 KB (1,196 words) - 13:02, 24 August 2015
• Elementary ODEs
  where $G:{\bf R}\to {\bf R}$ is some function and $q:C_1({\mathbb R})\to C_0({\mathbb R})$ is given by The growth is exponential (geometric), as expected. To verify, suppose $b:=[B,B+1]$. Then compute:
  16 KB (2,913 words) - 22:40, 15 July 2016
• Differentiation without limits: part 1
  We won't call it "rule" or "law" because it's about a specific function. Compare to: : for any [[differentiable function]] $f\cdot g$ wrt $x$.
  6 KB (1,004 words) - 16:00, 2 May 2011
• Calculus of sequences
  [[image:location as a function of time.png| center]] [[image:velocity as a function of time.png| center]]
  113 KB (18,425 words) - 13:42, 8 February 2019
• Multilinear forms
  In general, we consider a function: ...the dimension of the geometric object and the "degree" or "order" of this function.
  18 KB (3,325 words) - 13:32, 26 August 2013
• Geometry in calculus
  ...ing, deforming the $x$- or the $y$ axes won't change the monotonicity of a function but it will change its concavity. Concavity is an important concept that captures the shape of the graph of a function in calculus as well as the acceleration in physics.
  9 KB (1,604 words) - 18:08, 27 August 2015
• Limits: part 1
  ...complete data. Suppose we discover an exact formula. The altitude $y$ as a function of time, $x$, is \lim_{x\to 2} g(x) & = \lim_{x\to 2} (2 - x) \gets \text{ linear function }\\
  10 KB (1,609 words) - 16:13, 2 May 2011
• Calculus I -- Fall 2017
  ...near functions, power functions, polynomial functions, rational functions, exponential and logarithmic functions, and trigonometric functions. **The limit of a function at a point. One-sided limits. Continuity and the intermediate value theorem
  13 KB (2,075 words) - 13:35, 27 November 2017
• Calculus I -- Fall 2018
  ...near functions, power functions, polynomial functions, rational functions, exponential and logarithmic functions, and trigonometric functions. **The limit of a function at a point. One-sided limits. Continuity and the intermediate value theorem
  8 KB (1,184 words) - 17:55, 29 October 2018
• Calculus I -- Spring 2017
  ...near functions, power functions, polynomial functions, rational functions, exponential and logarithmic functions, and trigonometric functions. **The limit of a function at a point. One-sided limits. Continuity and the intermediate value theorem
  12 KB (1,803 words) - 20:50, 1 May 2017
• Calculus I -- Fall 2016
  ...near functions, power functions, polynomial functions, rational functions, exponential and logarithmic functions, and trigonometric functions. **The limit of a function at a point. One-sided limits. Continuity and the intermediate value theorem
  11 KB (1,671 words) - 23:11, 13 December 2016
• Exponential identity of functions
  The ''exponential identity of functions'': Another version of the identity is indeed ''exponential'':
  682 bytes (119 words) - 13:09, 29 June 2013
• College Algebra -- Fall 2016 -- final exam
  $\bullet$ '''2.''' A sketch of the graph of a function $f$ and its table of values are given below. Complete the table: $\bullet$ '''3.''' Find the implied domain of the function given by:
  2 KB (283 words) - 03:08, 6 November 2018
• Determinants of linear operators
  The goal is to find a function that determines whether $A$ is singular. It is called the ''determinant''. i.e., ''a linear function preserves addition''.
  19 KB (3,177 words) - 18:59, 10 October 2011
• Algebra of differential forms
  For example, we see below that $\phi = x^2 dx + xy dy$ is a [[linear function]] of $dx$ and $dy$, non-linear for $x,y$. Finally, a continuous function
  11 KB (1,947 words) - 18:14, 22 August 2015

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