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• First derivative test
  ...test''': classify the [[critical point]]s based on change of the sign of [[derivative|f'(a)]] [[Image:first second derivative test.jpg]]
  3 KB (438 words) - 19:02, 7 August 2010
• Chapter 4: First Derivative Test
  #REDIRECT [[First Derivative Test]]
  35 bytes (4 words) - 17:46, 8 July 2011

Page text matches

• The derivative
  ...in on the point. What do we see exactly? There are two possibilities. The first one is, we might see point connected by straight edges: <!--200-->[[image:second derivative is computed with a spreadsheet.png| center]]
  75 KB (13,000 words) - 15:12, 7 December 2018
• Differential calculus
  ==Monotonicity, extreme points, and the derivative== ...words, this is where the slope of the tangent line is zero. But that's the derivative of our function. From the ''Power Formula'', we have:
  84 KB (14,321 words) - 00:49, 7 December 2018
• 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 definitio Now, find the derivative of $x\cdot e^{x}$ with PR: above set
  49 KB (8,436 words) - 17:14, 8 March 2018
• Differentiation
  Furthermore, the derivative is defined as a limit. Unlike the limits we saw prior to derivatives, this ...unctions differentiable at a point is differentiable at that point and its derivative is equal to the sum of their derivatives; i.e., for any two functions $f,g$
  82 KB (14,116 words) - 19:50, 6 December 2018
• Calculus of chain maps
  First, $f$ has to be a function that takes nodes to nodes: ...sure the continuity of the resulting curve, we plot the nodes of the graph first and then attach edges to them. Therefore, we require from the edge function
  41 KB (7,344 words) - 12:52, 25 July 2016
• Metric tensor
  ...ay the cells are attached to each other affects the matrix of the exterior derivative: ...ere we start to need geometry is when we move from the first to the second derivative.
  42 KB (7,131 words) - 17:31, 30 November 2015
• Differential calculus of parametric curves
  But first we consider the issue that lies at the heart of calculus: the rate of chang The first one is defined just as the common limit but instead of $t$ approaching some
  130 KB (22,842 words) - 13:52, 24 November 2018
• Calculus I, the discrete version
  <!--150-->[[image:first derivative and Monotonicity.png| center]] The first idea is illustrated below:
  64 KB (11,521 words) - 19:48, 22 June 2017
• Introduction to discrete calculus
  To get from here to the values of the area, we need first to extend $s$, as a function, from single edges to their combinations, $1$- ==The derivative of a $0$-cochain is a $1$-cochain==
  40 KB (6,983 words) - 19:24, 23 July 2016
• Geometry
  ...ay the cells are attached to each other affects the matrix of the exterior derivative: ...ere we start to need geometry is when we move from the first to the second derivative.
  41 KB (6,928 words) - 17:31, 26 October 2015
• The gradient
  ...e output space. The first column consists of all parametric curves and the first row of all functions of several variables. The two have one cell in common; ...ore, every continuous function is integrable and, therefore, is somebody's derivative. In this sense, the arrow can be reversed.
  74 KB (13,039 words) - 14:05, 24 November 2018
• Calculus on cubical complexes
  *the names of the cells are given in the first row; ..., are organized into vector spaces, one for each degree. Let's review this first.
  36 KB (6,218 words) - 16:26, 30 November 2015
• Discrete forms
  *the names of the cells are given in the first row; ..., are organized into vector spaces, one for each degree. Let's review this first.
  35 KB (6,055 words) - 13:23, 24 August 2015
• Freshman's introduction to discrete calculus
  <!--150-->[[image:first derivative and Monotonicity.png| center]] ==The derivative==
  42 KB (7,443 words) - 14:18, 1 August 2016
• Forms in Euclidean spaces
  *first we plot the curve (green) which is the restriction of our function $\varphi ...ply the restriction of the continuous form to the set of integers, for the first argument:
  44 KB (7,778 words) - 23:32, 24 April 2015
• ODEs
  ...le ''ordinary differential equations (ODEs)'' with respect to the exterior derivative $d$ that have explicit solutions. ...e point of view and consider only the structures that we actually ''use''. First, since ring $R$ has no topology that we use,
  47 KB (8,415 words) - 15:46, 1 December 2015
• Metric complexes
  ...o each other affects the matrix of the boundary operator (and the exterior derivative): ...( u , v)= \langle u , v \rangle $. Then $p$ is linear with respect to the first variable, and the second variable, separately:
  35 KB (5,871 words) - 22:43, 7 April 2016
• Applications of differential calculus
  Warning: the method fails when it reaches a point where the derivative is equal to (or even close to) $0$. The most important use of the latter notation is in the definition of the derivative:
  59 KB (10,063 words) - 04:59, 21 February 2019
• Series
  We first approximate the function with a ''constant'' function: ...pproximate a function around the point $(1,1)$ with ''constant'' functions first; from those we choose the horizontal line through the point. This line then
  113 KB (19,100 words) - 23:07, 3 January 2019
• Applications of integral calculus
  ...realize that we could produce the same result if we take the data from the first spreadsheet, $\sum_i f(c_i)\cdot.1$, and them subtract the data for the new ...n sum of the new function in terms of areas. It's as if the rectangles are first aligned with $y=f(x)$, then cut from below with $y=g(x)$, suspended in the
  103 KB (18,460 words) - 01:01, 13 February 2019
• Several variables
  First, the ''sequences''. ...e time is in the first column progressing from $0$ every $.05$. The second derivative is in the next, $0$ and $-32$, respectively. In the next column, the initia
  76 KB (13,017 words) - 20:26, 23 February 2019
• Cubical calculus
  <center> ''a function on the right and its derivative is on the left''. </center> ...fferential forms. Then the form on the left is what we call the ''exterior derivative'' of the form on the right. Consequently, the theorem can be turned into a
  25 KB (4,238 words) - 02:30, 6 April 2016
• Functions of several variables: derivatives and integrals
  ==The derivative of a function of several variables== First let's look at the point-slope form of ''linear functions'':
  42 KB (6,904 words) - 15:15, 30 October 2017
• Functions of several variables
  ...lumn of this table. It is now time to move to the right. We retreat to the first cell because the new material does not depend on the material of Chapter 17 ...numerical functions. Hence the need for parametric curves. Similarly, the first cell of the second column has surfaces but not all of them because some of
  97 KB (17,654 words) - 13:59, 24 November 2018
• Tangent bundles
  ...arer when the dimension of the space is $2$ or higher. We use ''vectors''. First, as we just saw, the work of the force is $W = \pm F \cdot D$ if $F || D$, First, what is the set of all possible directions on a ''graph''? We've come to u
  16 KB (2,753 words) - 13:55, 16 March 2016
• Calculus of sequences
  First, imagine that our speedometer is broken. What do we do if we want to estima *after the first hour: $10,055$ miles;
  113 KB (18,425 words) - 13:42, 8 February 2019
• Derivative as a function
  ==Motion and the derivative== ...have three values of $g^{\prime}$. That a new function! It is called the ''derivative function''.
  9 KB (1,437 words) - 14:05, 7 October 2012
• Calculus exercises: part III
  Solution: The first two components indicate that for Solution: The first component is defined and continuous for
  46 KB (8,035 words) - 13:50, 15 March 2018
• Vector and complex variables
  .... Solve the new equation: $-x=-4$, or $x=4$. Substitute this back into the first equation: $(4)+y=6$, then $y=2$. The four coefficients of $x,y$ form the first table:
  113 KB (18,750 words) - 02:33, 10 December 2018
• Transformations
  .... Solve the new equation: $-x=-4$, or $x=4$. Substitute this back into the first equation: $(4)+y=6$, then $y=2$. Let's collect the data in tables first:
  46 KB (7,625 words) - 13:08, 26 February 2018
• Applications of ODEs
  *first, the one with the sides $p_n-x_n$ and $y_n-q_n$, and First, the characteristic polynomial is:
  50 KB (8,692 words) - 14:29, 24 November 2018
• Combinatorial cell maps
  ...result is used as a template to define cell maps. The difference is that, first, if $f(s)$ is attached to itself along its boundary but $s$ isn't, cloning ...h of these examples, an idea of a map $f$ of the circle/square was present first, then $f$ was realized as a chain map $g$.
  31 KB (5,330 words) - 22:14, 14 March 2016
• Differential equations
  Recall first that an ''augmented partition'' of an interval $[a,b]$ is a sequence of $n$ In the meantime, the derivative, if any, would satisfy the following:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Extrema of functions of several variables
  *solve [[derivative|$f'(x)$]] $= 0$ for $x$, (also find $x$'s for which $f'(x)$ does not exist) *[[second derivative test]]: classify the critical points based on the sign of $f' '(a)$. Note t
  9 KB (1,511 words) - 16:07, 17 August 2011
• PDEs
  ...of the ODE of population growth and decay -- with respect to the exterior derivative $d_t$ over time. This time, however, for each cell there are four adjacent *$d_t$ is the exterior derivative with respect to time (just the difference in ${\mathbb R}$); and
  44 KB (7,469 words) - 18:12, 30 November 2015
• Functions of several variables OLD
  First, we have replaces the absolute value with the [[norm]]. This simply reflect To define [[derivative]]s we need limits and for [[limits]] we need to understand better the [[top
  34 KB (5,636 words) - 23:52, 7 October 2017
• Vector fields
  The first metaphor for a vector field is a ''hydraulic system''. *$m$ is the mass of the first object;
  91 KB (16,253 words) - 04:52, 9 January 2019
• Integration
  But we don't recognize $\sin (x^{2})$ as the derivative of any function we know... *the derivative of the “inside” function is present as a factor.
  69 KB (11,727 words) - 03:34, 30 January 2019
• Diffusion equation
  ...of the ODE of population growth and decay -- with respect to the exterior derivative $d_t$ over time. For each cell there are four adjacent cells and four tempe *$d_t$ is the exterior derivative with respect to time (simply the difference in ${\mathbb R}$); and
  35 KB (5,917 words) - 12:51, 30 June 2016
• Systems of ODEs
  First, we assume that they have an unlimited food supply and reproduce in a manne First, we assume that the foxes have only a limited food supply, i.e., the rabbit
  63 KB (10,958 words) - 14:27, 24 November 2018
• Introduction to discrete calculus, continued
  First, $f$ has to be a function that takes nodes to nodes: ...o ensure continuity of the resulting curve, we plot the nodes of the graph first and then attach edges to them. If we discover that this is impossible, no r
  22 KB (3,661 words) - 13:12, 18 July 2016
• More metric complexes
  Given now a cell complex $K$ of an ''arbitrary'' dimension, the first step in the construction its dual $K^\star$ is to choose the ''dimension'' First we consider the case of $n=2$:
  21 KB (3,445 words) - 13:53, 19 February 2016
• Geometric complexes
  Given now a cell complex $K$ of arbitrary dimension, the first step in the construction its dual $K^\star$ is to choose the ''ambient dime First we consider the case of $n=2$:
  20 KB (3,354 words) - 17:37, 30 November 2015
• Discrete forms and cochains
  <center>''a function on the right and its derivative is on the left''. </center> ...as differential forms. The form on the left is what we call the ''exterior derivative'' of the form on the right.
  34 KB (5,619 words) - 16:00, 30 November 2015
• Integer-valued calculus
  First question: does the universe have a ''hole'' (or tunnel) in it? Is it possib We compute the rotor of the latter first:
  27 KB (3,824 words) - 19:07, 26 January 2019
• Topology Illustrated -- Index
  *[[derivative of a cell map|derivative of a cell map]] *[[Double Derivative Identity|Double Derivative Identity]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Partial differential equations
  Let's first review the material about the partial differences and difference quotients. ...p quickly cools down. At this point we start our simulation. These are the first two sheets: the permeability (zero around the edge of the cup) and the init
  53 KB (9,682 words) - 23:19, 18 November 2018
• Properties of the exterior derivative
  ==Leibniz formula, the product rule for exterior derivative== First, for $\varphi, \psi \in \Omega^0$, we have the familiar [[Product Rule]]:
  8 KB (1,539 words) - 18:17, 22 August 2015
• Limits: transition from discrete to continuous
  Then to understand limits of sequences in general, we need first to understand those of a smaller class: Then, by the first theorem, we have:
  51 KB (9,271 words) - 20:02, 8 September 2016
• Diffusion
  *$d_t$ is the exterior derivative with respect to time (just the difference since the dimension is $1$); and *$d_x$ is the exterior derivative with respect to location.
  39 KB (6,850 words) - 15:29, 17 July 2015
• Second derivative and the Laplacian
  ...is always trivial. Instead, one can define the "dualized" second exterior derivative as follows: *"dualize" the [[exterior derivative]] $df$,
  4 KB (608 words) - 13:13, 28 August 2015
• Derivative as a limit
  ''Derivative'' is the [[slope]] of the [[tangent line]]. ...' the tangent line? (Silly answer: Line the slope of which is equal to the derivative).
  5 KB (857 words) - 13:57, 25 May 2011
• The integral
  \text{the derivative }&\text{ the integral}\quad\\ First, we have a ''partition'' of $[a,b]$ into $n$ intervals of possibly differen
  66 KB (11,473 words) - 21:36, 19 January 2019
• Differentiation without limits: part 3
  \left(\frac{A}{B + C e^{x}}\right)^{\prime} & = ?? \text{Quotient rule first?} \\ # Derivative of $\sin$ at 0, find $(\sin x)^{\prime}$.
  5 KB (804 words) - 15:56, 2 May 2011
• Higher order votes
  We will restate some of the axioms. The first is without change. After all, the derivative of a monotonic function is either all positive or all negative.
  41 KB (6,942 words) - 05:04, 22 June 2016
• First derivative test
  ...test''': classify the [[critical point]]s based on change of the sign of [[derivative|f'(a)]] [[Image:first second derivative test.jpg]]
  3 KB (438 words) - 19:02, 7 August 2010
• Calculus 1: course
  ...ourse in introductory calculus. The main goal is some familiarity with the derivative and its applications. *[[Derivative as a limit]]
  2 KB (272 words) - 00:27, 25 September 2013
• Multilinear algebra
  First, what is the set of all possible directions on a graph? We've come to under First, observe:
  49 KB (8,852 words) - 00:30, 29 May 2015
• Hodge duality of differential forms
  First, it reveals a certain symmetry of the [[de Rham complex]]. Indeed, for $n$ ...$-forms): the duality operator matches the algebra but, since the exterior derivative is reversed, not the calculus. Compare also to [[Poincare duality]] of [[ho
  8 KB (1,072 words) - 17:59, 24 April 2013
• Vector functions
  To plot $f$, we list the input first - in the first two columns, then the corresponding output in the last two columns. Then we ==Tangent plane and the total derivative==
  28 KB (4,769 words) - 19:42, 18 August 2011
• Geometric Hodge duality
  First, we can simply specify the angles between the rods. Mathematically, this is ==Second derivative and Hodge duality==
  5 KB (867 words) - 13:24, 19 May 2013
• Applied Calculus -- Spring 2015
  **2.1 The Derivative and the Slope of a Graph *Chapter 3. Applications of the Derivative
  9 KB (1,141 words) - 16:08, 26 April 2015
• Elementary PDEs
  ...of the ODE of population growth and decay -- with respect to the exterior derivative $d_t$ over time. For each cell there are two adjacent cells and two tempera *$d_t$ is the exterior derivative with respect to time; and
  16 KB (2,843 words) - 21:41, 23 March 2016
• Parametric curves
  *[[derivative]], The last one deserves a special attention. Let's restate it first:
  34 KB (5,665 words) - 15:12, 13 November 2012
• Geometry in calculus
  First, we rely on [[limits]] that use the [[absolute value]]: ...ly topological ideas such as [[basis of topology]]. This also covers the [[derivative]].
  9 KB (1,604 words) - 18:08, 27 August 2015
• Elementary ODEs
  ...differential equations (ODEs) of cochains'' with respect to their exterior derivative $d$. We choose a few simple examples that have explicit solutions. *the first derivative $f'$ instead of the exterior derivative, and
  16 KB (2,913 words) - 22:40, 15 July 2016
• Discrete functions
  ...may come from looking occasionally at your odometer. Suppose also that the derivative, $f'$, of this function -- representing the velocity of your car -- is also ...with all the functions being discrete, there is no differentiation or the derivative in the conventional sense. Can we ever establish such a relation between tw
  15 KB (2,532 words) - 12:21, 11 July 2016
• Calculus by Rogawski
  3.1 Definition of the [[Derivative]] 3.2 The [[Derivative as a function|Derivative as a Function]]
  6 KB (634 words) - 16:38, 1 March 2013
• Social choice
  We will look into the first two options as they are subject to the algebra we have developed in this ch ...to find the outcome vote. The second option is to tally at the end of the first day, then, separately, at the end of the second, and then add the two outco
  47 KB (8,030 words) - 18:48, 30 November 2015
• Calculus as a part of topology
  The early calculus is about the derivative, i.e., the rate of change of a function. This doesn't seem like a part of T These structures also interact with algebra leading the way. First, we can rephrase the last three questions as:
  13 KB (2,233 words) - 14:41, 20 February 2015
• Discrete parametric curves
  First, we know what it is in ${\bf R}^2$. *derivative,
  32 KB (5,426 words) - 21:57, 5 August 2016
• Fantasy math
  *The derivative of the composition is the composition of the derivatives. ...nd $y=g(x)=x^2$. From the same bad place comes plotting a function and its derivative (or its inverse) on the same $xy$-plane, or writing $\partial f/\partial x=
  8 KB (1,196 words) - 13:02, 24 August 2015
• Gradient
  How do we understand the [[derivative]] of [[functions of several variables]]? *The [[directional derivative]] $\nabla_e f(a)$ is a number for each $e$, $||e||=1$, there are infinitely
  6 KB (962 words) - 15:45, 17 August 2011
• Vector calculus: exercises
  Solution: The first two components indicate that for Solution: The first component is defined and continuous for
  16 KB (2,457 words) - 02:17, 22 August 2011
• Using derivative to study concavity
  Recall how we are [[using derivative to study monotonicity]]. ...owth of the derivative has slowed down! But the growth is described by the derivative!
  5 KB (854 words) - 01:30, 16 July 2011
• Differentiation without limits: part 4
  ...der $y = (1 + 2x)^{2}$. This is a composition of 2 functions. What's its [[derivative]]? \end{aligned} \right\} \qquad \text{How do we combine these to get the derivative of the whole?}
  7 KB (1,258 words) - 14:32, 1 May 2011
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  ...in a [[cell complex]] in terms of $(k-1$-cells, you also know the exterior derivative of all discrete [[differential forms]] ([[co-chain]]s). So, you know calcul ...he [[cell complex|complex]] itself''. In other words, the structure of the derivative operator depends on the [[topology|topological structure]] of the space --
  11 KB (1,663 words) - 16:03, 26 November 2012
• Wave equation
  First, we consider the ''spatial variable'', $x\in {\bf Z}\times {\bf Z}$. We thi Note that for a constant $k$, we are dealing with the second derivative of the $0$-form $u$ with respect to space:
  10 KB (1,775 words) - 02:40, 9 April 2016
• Integral theorems of vector calculus
  '''Theorem.''' Given a [[vector field]] $F = ( p, q )$ with continuous [[derivative]]. Then '''Theorem.''' Suppose vector field $F = ( p, q )$ has continuous derivative. Then
  16 KB (2,752 words) - 14:18, 28 December 2012
• Calculus and algebra vs topology
  Both concepts are reduced to the minimum here. The calculus is reduced to [[derivative|differentiation]] and the [[topology]] of $R$ is reduced to the number $m$! where the derivative operator takes continuously differentiable functions to continuous ones.
  4 KB (598 words) - 21:26, 8 February 2013
• Treat calculus discretely!
  *after the first hour: $10,055$ miles; *distance during the first hour: $10,055-10,000=55$ miles;
  8 KB (1,196 words) - 12:02, 4 July 2018
• Line integrals
  Suppose $X=X(t)$ is a parametric curve on $[a,b]$. The first is the (component-wise) ''integral of the parametric curve'': ...y field (it is explained in this section). The main difference between the first and the rest is that the parametric curve isn't the ''integrand'' (and the
  12 KB (2,194 words) - 14:37, 5 December 2017
• Calculus I -- Fall 2017
  *Goals: good understanding of limits, the derivative and the integral, fluent differentiation. *[[The derivative]]
  13 KB (2,075 words) - 13:35, 27 November 2017
• Discrete calculus article
  ..., however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is se <ref>{{cite book|last=Levy|first=H.|author2=Lessman, F.|title=Finite Difference Equations|year=1992|publishe
  27 KB (4,329 words) - 16:02, 1 September 2019
• Affine approximation
  ...linear map evaluated at $x-a$. This linear map $L_a$ is called the ''total derivative of $f$ at $x = a$''. Then, the total derivative
  7 KB (1,162 words) - 03:25, 22 August 2011
• Calculus 1: final exam
  ...the derivative of a function at point $a$. (b) Use part (a) to compute the derivative of $f(x)=x²+3$ at $a=1$. 6. The graph of function $f$ is given below. Sketch the graph of the derivative $f′(x)$ in the space under the graph of $f$. Identify all important point
  2 KB (349 words) - 20:21, 13 June 2011
• Calculus on chains
  where $d$ is the [[discrete exterior derivative]]. The conclusion is based on the shortest version of the [[Stokes Theorem] ==Main properties of exterior derivative==
  4 KB (556 words) - 14:03, 30 March 2013
• Using derivative to study monotonicity
  We need to know the sign of the derivative on each. '''Corollary.''' When two have the same derivative, you can get the graph of one from the other by a vertical shift.
  4 KB (624 words) - 00:56, 16 July 2011
• Integration of differential forms of degree 2
  ...f the operation [[inverse]] to differentiation, in our case the [[exterior derivative]]. We will have to develop some background first, i.e., the [[Stokes Theorem]].
  15 KB (2,341 words) - 20:53, 13 March 2013
• Examples of differential forms
  We will start our exploration with familiar idea of the [[derivative]] from [[calculus 1:_course|calculus 1]]. Consider the two notations for the [[derivative]] at $a$ of $f$:
  5 KB (959 words) - 13:15, 12 August 2015
• Continuous vs discrete differential forms
  It's like the second line is a grainy image of the real life in the first line. How is this image made and how much information does it preserve. <center>''sampling a function and its derivative produces two discrete functions that don't satisfy the [[Fundamental Theore
  9 KB (1,483 words) - 13:54, 13 April 2013
• Convergence of the discrete to the continuous
  ...we acquire first the derivative $f'$ of $f$, and then we find the exterior derivative (a $1$-form) $dg$ by integrating $f'$ on the segments: '''Exercise.''' Show that, in this case, all the values of the derivative $f'$ of $f$ are the limits of sequences of values of $g_k'$, under the assu
  21 KB (3,664 words) - 02:02, 18 July 2018
• Directional derivative
  These two cases lead to the concept of [[partial derivative]]s. then the [[derivative]] is the rate of change of $y$ with respect to $x$. We will use this concep
  4 KB (715 words) - 20:12, 28 August 2011
• Discrete Calculus. An Introduction
  ...alculus of differential forms. The ''continuous'' counterpart is developed first because, typically, it is not a part of a calculus course. Meanwhile, we wa ##[[Exterior derivative]]
  4 KB (466 words) - 19:07, 8 July 2014
• Advanced Topology -- Spring 2013
  <TR> <TD><center>Derivative vs boundary</center></TD></TR> #[[Exterior derivative]]
  5 KB (725 words) - 14:49, 8 May 2013
• Wedge product of continuous forms
  The first thought is that we just multiply them: First, it has to be linear on $(v_x,v_y)$ and linear on $(v'_x,v'_y)$. The regul
  14 KB (2,417 words) - 18:16, 22 August 2015
• Modelling with discrete vector fields and forms
  where $x$ is still a $0$-form, but $dx$ its [[exterior derivative]], a $1$-form, and $dt$ the constant $1$-form. Suppose we discretize ''time'' first. Then $x$ is simply a function of discrete variable. And so can $dx$, as a
  9 KB (1,561 words) - 16:06, 27 August 2015
• Linear functions in Euclidean space
  First let's address some terminology... (The functions have same derivative though.)
  23 KB (3,893 words) - 04:43, 15 February 2013
• Modeling with discrete exterior calculus
  where $d_{x}$ denotes an exterior derivative with respect to $x$ and $\star$ denotes the use of the Hodge Duality. ...x}/4$ is shown above. The axes have been modified so that it resembles the first lattice shown. Here is another image as the temperature distributes itself
  31 KB (5,254 words) - 17:57, 21 July 2012
• Excel simulations
  ...[ video] [http://inperc.com/files/simulations/pva.xlsx spreadsheet] [[The derivative#A ball is thrown...|background]], [[Applications_of_ODEs#A_cannon_is_fired. ...video] [http://inperc.com/files/simulations/cannon.xlsx spreadsheet] [[The derivative#A ball is thrown...|background]]
  5 KB (822 words) - 00:38, 3 September 2018
• Calculus I -- Fall 2016
  *Goals: good understanding of limits, the derivative and the integral, fluent differentiation *[[The derivative]]
  11 KB (1,671 words) - 23:11, 13 December 2016
• Calculus I -- Spring 2017
  *Goals: good understanding of limits, the derivative and the integral, fluent differentiation *[[The derivative]]
  12 KB (1,803 words) - 20:50, 1 May 2017
• Christopher Means
  The continuous wave equation can be expressed as the second time derivative of f equal to some constant squared ...fference of $f$ at time $t$ and $t-1$, or $t+1$ and $t$. The second order derivative, which is what the wave equation uses, is a difference of differences. It
  6 KB (1,025 words) - 23:41, 15 July 2012
• Critical point
  - solve [[derivative|f′(x)]] = 0 for x, - 2nd derivative test: classify the critical points based on the sign of f''(a).
  888 bytes (158 words) - 17:45, 7 August 2010
• Discrete differential forms
  ...i.e., we only know its values at, say, 0, 1, 2, 4, 5... Suppose that the [[derivative]], $f'$, of the function is also sampled: Then it would make sense if $g'$ was the [[derivative]] of $g$, in some way.
  10 KB (1,471 words) - 12:50, 12 August 2015
• Cochains on graphs
  The first crucial difference is that we can ''multiply'' the elements of the latter s ...s what we need to consider as $th$ makes sense. Then we simply require the first diagram to be commutative or, which is the same thing, that the outputs of
  15 KB (2,523 words) - 18:08, 28 November 2015
• Calculus on graphs
  The first crucial difference is that we can ''multiply'' the elements of the latter, ...consider -- $th$ makes sense and $ht$ doesn't. Then we simply require the first diagram to be commutative or, which is the same thing, that the outputs of
  16 KB (2,578 words) - 00:14, 18 February 2016
• Second derivative test
  '''Second derivative test'': [[First derivative test]]: classify the critical points based on change of the sign of f'(a)
  495 bytes (85 words) - 17:48, 7 August 2010
• Modern Thermodynamics by John Denker
  *"The first law of thermodynamics states that energy obeys a local conservation law. ...the word comes from: this k-form is (or is not) ''exactly'' the [[exterior derivative]] of some (k-1)-form.
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• Computing integrals
  We don't recognize $\sin (x^{2})$ as the derivative of any function we know. Conclusion: we can integrate compositions when ''the derivative of the "inside function" is present as a factor'':
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• Calculus II -- Spring 2018
  *Prerequisites: excellent algebra skills, good understanding of limits, the derivative, and the integral, fluent differentiation -- [[Calculus I -- Fall 2017|Calc ...r [http://inperc.com/files/pva.xlsx download], some explanations are [[The derivative#A ball is thrown...|here]].
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• Finite differences
  ==First order== The limits of these as $h \rightarrow 0$ give you the [[derivative]] of $f$, if it exists.
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• Forms
  ...the displacement vector. Then this is what we know about the work of $F$. First, as we just saw, the work of the force is First, what is the set of all possible directions on a graph? We've come to under
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• Preview of calculus: part 2
  He would first try to apply his common sense. Later, we will use the derivative. Now we use the fact that this is a [[parabola]].
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• Motion planning in robotics
  ...bf T}^2)={\bf R}\times{\bf R}$ is generated by the two joints but only the first one is mapped to the the generator of $H_1(A)={\bf R}$. The second joint do with the rods (first red, second green) perpendicular to each other, the configuration space and
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• The algebra of oriented cells
  These “scalars” are chosen from some collection $R$. What kind of set is $R$? First, we need to be able to ''add chains'', But first we observe that, if the intervals overlap, the property still makes sense,
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• Inner product spaces: part 1
  With this you can introduce [[limit]]s, [[continuity]], [[derivative]], [[integral]], the rest of [[calculus]]. So, we have calculus in ${\bf R} First, we know what it is in ${\bf R}^2$.
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• Matrices as functions
  ...\colon {\bf R}^2 \rightarrow {\bf R}^2$, as vector spaces. Elements in the first ${\bf R}^2$ are considered to be $\left[ *[[derivative]],
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• Exterior derivative with Excel
  The [[exterior derivative]] in dimensions 1 and 2 can be computed with Excel. [[image:exterior derivative in dimension 1 Excel.png|center]]
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• Closed and exact forms
  '''Theorem (Topological property of exterior derivative).''' If $\varphi$ is a $C^2$-form in $\Omega ^k({\bf R}^n)$, then $d_{k+1}( ...twice continuously differentiable. Then, using the definition of exterior derivative and its formula for $0$-forms we have:
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• Calculus Two by Flanigan and Kazdan
  *6.2 [[Directional derivative]]s 333 *6.6 The World of First Derivatives 368
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• Calculus exercises: advanced
  *Use an appropriate change of variables to find the area of the first-quadrant region bounded by the curves $xy=1,\ xy=2$ and $xy^2=2,\ xy^2=4$. *Find the matrix of the total derivative of $F(x,y)=(x\sin y,x-y)$ at $(1,0).$
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• Isotropy in numerical PDEs
  ...unknown per point, and writes one equation per point. At each point, the [[derivative]]s of the unknown are replaced by finite differences via [[Taylor series]] *First, the coefficients involved in the equation may be location- (e.g. in the ca
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• Discrete calculus: contributors
  ...logy]]. Therefore, the credit for the creation of discrete calculus should first go to the following individuals (roughly 1850 - 1950): ...mplex|dual triangulation]]), [[Poincare duality]], [[Poincare lemma]], the first proof of the general [[Stokes Theorem]], and a lot more;
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• The algebra of cubical cells
  To approach this question, let's try to answer another first: These “scalars” are chosen from some collection $R$. What kind of set is $R$? First, we need to be able to ''add chains'',
  46 KB (7,844 words) - 12:50, 30 March 2016
• De Rham cohomology
  *The [[exterior derivative]] is a [[linear operator]] $d_0 \colon \Omega^0(R) \rightarrow \Omega^1(R)$ *The [[exterior derivative]] is a [[linear operator]] $d_1 \colon \Omega^1(R) \rightarrow \Omega^2(R)$
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• Parametric curves: exercises
  Parametrization of the first edge is: ...</sub>(s) ≠ g′₁(s), etc at the points of gluing, so there is no derivative here. For example, vompute g′( √2 ):
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• Derivative and integral: Fundamental Theorem of Calculus
  ...hat, we have to go all the way back to the definition of the derivative ([[derivative as a limit]]). Find the position function first: $F(x)$, then
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• Spreadsheets
  [[Exterior derivative]]: [[image:Exterior derivative in dimension 1 Excel.png|center]]
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• Differentiation formulas for exterior derivative
  So, the [[exterior derivative]] is linear, of course: ...$g_1$ a $1$-[[chain map]] generated by $g$ (Note: $g_1$ is, in a way, the derivative of $g=g_0$).
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• The algebra of cells
  We know how to compute the derivative of the product of, say, three functions. We simply apply the product rule t ...olved and each contains all three, except one of them is replaced with its derivative. Applying the rule for boundaries above will have the exact same effect. Th
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• Divergence
  ...ose $F = ( p, q )$ is the velocity field for the flow. The idea is that we first break the region into small pieces and answer the question for each and the ...em of Calculus]]. Thus, the divergence is simply a generalization of the [[derivative]].
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• Integration of differential forms of degree 0 and 1
  Answer: the derivative $p'$. Let's review [[line integral]]s ([[Calculus 3: course|calc 3]]) first.
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• Calculus 1: midterm 1 solutions
  First the plane descends fast, then slows down and lands safely. From the definition, compute the derivative of $f(x) = x^{2} + 1$ at $a = 2$.
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• Khan Academy
  ...e the derivative of a specific function than to understand the idea of the derivative. The result is a parade of computational examples that neatly fit into the
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• Calculus III -- Fall 2017
  *Prerequisites: excellent algebra skills, good understanding of the derivative and the integral, fluent differentiation and integration. ...r [http://inperc.com/files/pva.xlsx download], some explanations are [[The derivative#A ball is thrown...|here]].
  10 KB (1,596 words) - 13:34, 27 November 2017
• Linear operators: part 3
  Our interpretation: the derivative is a linear operator. ...arify, suppose $f(x,y) = (x^2-y,xy)$. What is $f'$? It's made of [[partial derivative]]s:
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• Calculus I -- Fall 2018 -- final
  $\bullet$ '''4.''' (a) Analyze the first and second derivatives of the function $f(x)=x^4-2x^2$. (b) Use part (a) to ...The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=0,3$, and $5$.
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• Group theory: final exam
  ...p-theoretic interpretation of the relation between the derivative and anti-derivative. #(a) State the First Isomorphism Theorem. (b) Prove that the isomorphism is well defined. (c) Gi
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• Chains vs cochains
  *the names of the cells are given in the first row, ...s or differential forms. Cochains (differential forms) need the [[exterior derivative]]. The former decreases the dimension/degree while the latter increases.
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• Guide for contributors
  ...n the text, such as <nowiki>[[</nowiki>derivative<nowiki>]]</nowiki> for [[derivative]], whenever possible but no more than one link to the same page per page. *Avoid capitalization in titles and elsewhere beyond the first word.
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• Closedness and exactness of 1-forms
  ...actness problem". Given a continuous function $f$, it is exact if it's the derivative of someone: We deal with closed and exact $1$-forms in ${\bf R}^1$ first.
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• Codifferential
  ...k(M)\rightarrow C^{k-1}(M)$$ is the operator [[adjoint]] of the [[exterior derivative]] (aka the differential) Since the exterior derivative satisfies $d^2=0$, the codifferential has the sameproperty. Indeed:
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• Exponential models
  We can restate this in calculus terms: ''the derivative of $y$ is proportional to $y$''. Or: Let's estimate first by assuming that decay is linear.
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• Cochain complexes and cohomology
  First, which of these forms are closed? They should have "horizontal difference - Second, what $1$-forms are exact? Here is a $0$-form and its exterior derivative:
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• Differential Equations -- Spring 2017
  '''MTH 335 - Differential Equations.''' First and second-order ordinary differential equations. Applications include vibr ...r [http://inperc.com/files/pva.xlsx download], some explanations are [[The derivative#A ball is thrown...|here]].
  9 KB (1,360 words) - 15:10, 1 June 2017
• Algebra of discrete differential forms
  First, their ''addition''. The sum $\varphi + \psi$ is a form of degree $k$ too a First the [[cochain]] representation:
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• Binary watershed
  <p>First, for each pixel compute the distance to the nearest white pixel. This is ca ...y grow the particles. The dilation, however, has to have two restrictions. First, the particles aren’t allowed to grow beyond the original set of black pi
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• Calculus III -- Spring 2015 -- final exam
  $\bullet$ '''2.''' (a) Give the definition of the curvature (as a certain derivative). (b) Use the definition to compute the curvature of a circle of radius $R$ ...he vector field $F=< 2,1 >$ across the part of the circle that lies in the first quadrant.
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• The algebra of oriented cubes
  These “scalars” are chosen from some collection $R$. What kind of set is $R$? First, we need to be able to ''add chains'', ...this cell. Then, indeed, there will be $s$ coming from the boundary of the first cell and $-s$ from the boundary of the second.
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• More about manifolds
  First, observe that We know that this is the first option ahead of time if we compute the [[derivative]] at $x=a$ with $f(a)=b$ and
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• Vector calculus: midterm
  #Let $f(t)=(t^3-3t,t^2)$. (a) Find the derivative $f'$ of $f$. (b) Use $f'$ to plot the parametric curve $f$. ...x-y+2z=0$$ $$-2(x-2)+2(y+2)-4(z+2)=0$$ intersect, a student multiplied the first one by $(-2)$ and then added the result to the second. He got $0=0$. Explai
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• Gestalt and computer vision
  ...preted as the assumed [[derivative|differentiability]] (smoothness) of the first curve...
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• Gray scale watershed
  ...ence becomes a problem when the image contains both kinds on features. The first and second images below are what watershed should be theoretically and the ...To determine the proper amount of smoothing, you need to analyze the image first…
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• Identities of vector calculus
  ...bf R}^2$ diagram and examine the operators corresponding to the [[exterior derivative]] $d$. First,
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• Chapter 4: First Derivative Test
  #REDIRECT [[First Derivative Test]]
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• Dot product
  With this you can introduce [[limit]]s, [[continuity]], [[derivative]], [[integral]], the rest of [[calculus]]. So, we have calculus in ${\bf R} First, we know what it is in ${\bf R}^2$.
  2 KB (410 words) - 15:09, 9 June 2012
• Orientation of manifolds
  First, none of these should be equal to 0. ...point it becomes clear that this is about [[linear algebra]]. Indeed the [[derivative]] $T'$ is a [[linear operator]],
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• Calculus II -- Fall 2014
  *Prerequisites: excellent algebra skills, good understanding of limits, the derivative, and the integral, fluent differentiation -- [[Calculus 1: course|Calculus *week 1, 8/25: ''First HW called Review_set is due Friday 8/29 at midnight.'' Quiz
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• PageRank as a flow
  where $d$ stands for the [[exterior derivative]]. If you've read the rest of the [[book]] and you know how to make your first billion!
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• Calculus exercises: part II
  First, we consider the constant approximation $T_0$ which produces the value of $ [[image:estimate derivative from graph.png| center]]
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• Properties of integrals of differential forms
  First, what are the requirements on $p$, $q$? *Their [[derivative]]s are non-zero, $p' \neq 0$, $q' \neq 0$. ('''Exercise:''' why do we need
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• Change of variables for differential forms
  One thing we can observe is what happens under $F$ to the [[exterior derivative]] of the forms. For the $0$-forms: For question (c), let's consider $0$-forms only first, i.e., functions.
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• Rates of change
  We will consider a new examples of applications of the [[derivative]] as the rate of change of a variable. However, let's recall ''the'' exampl I. First name things, all of them:
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• Peter Saveliev
  ...? For now, I just picked enough material for these three chapters from the first 3 volumes of Calculus Illustrated. [https://www.dropbox.com/s/k4hdqqixming **Chapter 4 The Derivative [https://www.dropbox.com/s/q7ur2pmy259popd/ch4.pdf?dl=0 pdf]
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• Real analysis: course
  ...ester course in real analysis (aka advanced calculus). Certain facts about derivative, integral, the real number system, continuity and differentiability of func Text: ''A First Course in Real Analysis'' by Protter and Morrey (Chapters 2-5,9).
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