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• Differential calculus
  ...umn. One can also watch the sign of the differences of $A$; the values are increasing when the ''difference'' is positive! These are the points where the differe Furthermore, examining the graph reveals that the function is increasing where the slopes of the tangent lines are positive and decreasing where the
  84 KB (14,321 words) - 00:49, 7 December 2018
• Classes of functions
  When an object is moving up, we also say that its altitude is ''increasing'' and then, when its falling down, the altitude is ''decreasing''. We apply ...ning is very clear. However, both are imprecise. Even though we understand increasing functions as ones with graphs rising and decreasing functions as one with g
  143 KB (24,052 words) - 13:11, 23 February 2019
• Calculus exercises: part I
  *how fast is the level increasing? * monotonicity (increasing/decreasing),
  49 KB (8,436 words) - 17:14, 8 March 2018
• Calculus of sequences
  As you can see, the behavior varies even within these two categories: increasing and decreasing. [[image:increasing and decreasing sequences.png| center]]
  113 KB (18,425 words) - 13:42, 8 February 2019
• Precalculus
  *if $x$ is increasing, then $y$ is increasing; 1&1&\text{ increasing?}\\
  100 KB (16,148 words) - 20:04, 18 January 2017
• Calculus I, the discrete version
  '''Definition.''' A node function $f$ is called ''increasing on interval'' $[a,b]$ if ...rse, the function is increasing (or decreasing) on the interval when it is increasing (or decreasing) on each edge within the interval:
  64 KB (11,521 words) - 19:48, 22 June 2017
• Preview of calculus: part 2
  == Growth: Increasing/Decreasing Behavior == Function \( y=f(x) \) is ''increasing on interval'' \( (a, b) \) if for any given \( x_{1}, x_{2} \) in \( (a, b)
  19 KB (2,850 words) - 15:04, 19 March 2011
• Functions
  '''Example.''' An even simpler case is a list of numbers arranged in increasing order; then the task is easy: ...een calculated for you so that you can see patterns; for example, with $x$ increasing we see that
  151 KB (25,679 words) - 17:09, 20 February 2019
• Limits and continuity
  *increasing the pressure continuously in a closed container -- until it explodes, *increasing the temperature continuously of a piece of ice -- until it melts.
  107 KB (18,743 words) - 17:00, 10 February 2019
• Differential equations
  f(t,y)>0\ \Longrightarrow\ y \text{ is increasing};\\ Furthermore, every solution $y$ is decreasing (or increasing) throughout its domain. If we can demonstrate existence, every solution is
  64 KB (11,426 words) - 14:21, 24 November 2018
• Plotting the graph of a function
  #: * [[monotonicity|Increasing/decreasing]] ''Wrong:'' <del>So, $f$ increasing, on domain of $f$:</del>
  7 KB (1,132 words) - 17:34, 16 July 2011
• Preview of calculus: part 1
  ...reas by filling them with increasing number of rectangles or triangles and increasing the number of sides with each step. We can then calculate the area by calcu # If \( x \) is increasing, then \( y \) is increasing.
  10 KB (1,532 words) - 00:07, 2 May 2011
• Sequences and their limits
  ...ce Theorem).''' If a sequence is bounded and monotonic, i.e., it is either increasing, $a_n\le a_{n+1}$ for all $n$, or decreasing, $a_n\ge a_{n+1}$ for all $n$, is increasing. We have:
  64 KB (10,809 words) - 02:11, 23 February 2019
• Using derivative to study monotonicity
  This is helpful but what about ''[[increasing]]/decreasing'' behavior? m > 0 & \Rightarrow f \text{ increasing} \\
  4 KB (624 words) - 00:56, 16 July 2011
• Freshman's introduction to discrete calculus
  '''Theorem (Monotonicity).''' A node function is increasing (decreasing) on an interval if and only if its differential is positive (ne '''Theorem (Monotonicity).''' A node function is increasing (decreasing) on an interval if and only if its derivative is positive (nega
  42 KB (7,443 words) - 14:18, 1 August 2016
• Preview of calculus: part 3
  Indeed term "exponential growth" makes sense: \( a ^{x} \) is ''[[increasing]]'' for \( a > 1 \). a^{x} & \textrm{ is increasing if } a > 1 \\
  17 KB (2,498 words) - 15:06, 19 March 2011
• Several variables
  Second, suppose that the price of sugar is increasing and then decreasing: What kind of dependencies are these? Increasing prices of the ingredients in creases the cost and ultimately the price of t
  76 KB (13,017 words) - 20:26, 23 February 2019
• Vector and complex variables
  Now, increasing the value of $q$ makes the graph of $y=f(x)$ shift upward and, eventually, ...o ''the signs of the real parts of their roots''. The signs will determine increasing and decreasing behavior of certain solutions. Once again, these are the pos
  113 KB (18,750 words) - 02:33, 10 December 2018
• Introduction to discrete calculus
  '''Theorem (Monotonicity).''' A $0$-cochain is increasing (decreasing) on an interval if and only if its exterior derivative is posit '''Theorem (Monotonicity).''' A $0$-cochain is increasing (decreasing) on an interval if and only if its derivative is positive (nega
  40 KB (6,983 words) - 19:24, 23 July 2016
• Differentiation
  As the width and the depth are increasing, so is the area of the rectangle. But the increase of the area cannot be ex *increasing slopes = tangents rotate counter-clockwise.
  82 KB (14,116 words) - 19:50, 6 December 2018
• Applications of integral calculus
  '''Exercise.''' Prove that if the density of the rod is strictly increasing (or decreasing), its center of mass cannot be in the center. '''Exercise.''' Find the center of mass of a rod with a linearly increasing density.
  103 KB (18,460 words) - 01:01, 13 February 2019
• Calculus 1: exercises
  *Sketch the graph of a function f that has the following property: f is increasing and concave down on $[-1,1]$. *$f′(x)=(e^{x})².$ (a) On what intervals, if any, if f increasing? (b) On which intervals, if any, is f concave down? (c) Sketch the grap
  3 KB (435 words) - 19:23, 13 June 2011
• Parametric complexes
  ...mplex, one considers all thresholds and all possible cell complexes. Since increasing threshold $r$ enlarges the corresponding complex, we have a sequence of com ...subsections: (a) the Vietoris-Rips construction of the circle, and (b) the increasing resolution of the triangle.
  45 KB (7,255 words) - 03:59, 29 November 2015
• Calculus of chain maps
  '''Theorem (Monotonicity).''' A $0$-chain map is increasing (decreasing) on an interval if and only if its exterior derivative is posit '''Theorem (Monotonicity).''' A $0$-chain map is increasing (decreasing) on an interval if and only if its derivative is positive (nega
  41 KB (7,344 words) - 12:52, 25 July 2016
• Metric tensor
  ...it is sufficient to know the sign of the derivative to distinguish between increasing and decreasing behavior. Therefore, this behavior depends only on the topol ...g, the sign of the exterior derivative will tell us the difference between increasing and decreasing behavior. But the derivative only uses the topological prope
  42 KB (7,131 words) - 17:31, 30 November 2015
• Functions of several variables
  ...variable. This is how the concavity with respect to $x$ is increasing with increasing $y$:
  97 KB (17,654 words) - 13:59, 24 November 2018
• Geometry
  ...it is sufficient to know the sign of the derivative to distinguish between increasing and decreasing behavior. Therefore, this behavior depends only on the topol ...g, the sign of the exterior derivative will tell us the difference between increasing and decreasing behavior. But the derivative only uses the topological prope
  41 KB (6,928 words) - 17:31, 26 October 2015
• Convergence of the discrete to the continuous
  But does the latter approximate the former? Will increasing the “resolution” of the discretization allow us to recover the original ...his problem the same way one would deal with other accuracy problems -- by increasing the resolution of the image. In fact, a curve can be approximated by a digi
  21 KB (3,664 words) - 02:02, 18 July 2018
• Differential calculus of parametric curves
  ...n the curve is oriented, i.e., its direction is indicated, and $t=g(s)$ is increasing. Now assuming that $s$ is increasing with respect to $t$ (same direction!), we have a convenient way to describe
  130 KB (22,842 words) - 13:52, 24 November 2018
• Applied Calculus -- Spring 2015
  **3.1 Increasing and Decreasing Functions **3.1 Increasing and Decreasing Functions
  9 KB (1,141 words) - 16:08, 26 April 2015
• Calculus 1: midterm 2
  ...ncreasing at a rate of 5 cm²/sec. At what rate is the radius of the circle increasing when the area is 2 cm?
  991 bytes (164 words) - 20:21, 13 June 2011
• Precalculus exercises
  *Prove, from the definition, that the function $f(x)=x^2+1$ is increasing for $x>0$. ...n below. Describe its behavior the function using words “decreasing” and “increasing”.
  17 KB (2,933 words) - 19:37, 30 July 2018
• Calculus 1: midterm 2 solutions
  ...t a rate of 5 cm²/sec. At what rate is the radius of the circle increasing when the area is 2cm?
  3 KB (458 words) - 04:13, 21 May 2011
• Preview of calculus
  ...reas by filling them with increasing number of rectangles or triangles and increasing the number of sides with each step. We can then calculate the area by calcu
  4 KB (703 words) - 14:34, 9 September 2016
• Calculus 1: test 2
  ...reasing at a rate of $5$ cm²/sec. At what rate is the radius of the circle increasing when the area is $2$ cm?
  976 bytes (164 words) - 20:22, 13 June 2011
• Series
  '''Proof.''' First, the sequence of partial sums of this series is ''increasing'': ...partial sums of a series $\sum a_n$ with non-negative terms, $a_n>0$, is ''increasing'':
  113 KB (19,100 words) - 23:07, 3 January 2019
• Calculus II -- Fall 2018 -- midterm
  ...creasing or decreasing; (b) $b_n\to +\infty$ as $n\to \infty$ but it's not increasing.
  1 KB (246 words) - 19:10, 31 October 2018
• Social choice
  ...usly increases or decreases in both locations. Such a function can only be increasing or decreasing, i.e., monotonic. ...es are computed and displayed on the right. The scores for $A, ..., J$ are increasing in the obvious way: from $0$ to $.1$. In particular, $F$ is $\#$5.
  47 KB (8,030 words) - 18:48, 30 November 2015
• Graph representation of topology of color images
  ...mes [0,M] → [0,255]$ is a [[gray scale image]]. Suppose also that $P$ is [[increasing]] with respect to the partial order:
  4 KB (671 words) - 19:56, 30 October 2012
• Rating votes
  ...y increases or decreases in both locations. Such a function can only be an increasing or decreasing one, i.e., monotonic.
  9 KB (1,553 words) - 06:12, 22 June 2016
• Computational science training: 2011 projects
  In the order of increasing math background required...
  6 KB (794 words) - 00:56, 1 June 2012
• Calculus exercises: part III
  *Sketch a curve on the plane the curvature of which is: (a) increasing, (b) decreasing, (c) constant non-zero, (d) zero.
  46 KB (8,035 words) - 13:50, 15 March 2018
• Lengths of curves
  ...his problem the same way one would deal with other accuracy problems -- by increasing the resolution of the image. In fact, a curve can be approximated by a digi
  13 KB (2,113 words) - 20:21, 7 February 2013
• Operations on functions
  ...'t a function! To make sure, it's a good idea to arrange the inputs in the increasing order. Then we clearly see the conflict: $f^{-1}(1)=0$ and $f^{-1}(1)=3$. T
  142 KB (23,566 words) - 02:01, 23 February 2019
• Higher order votes
  ...es are computed and displayed on the right. The scores for $A, ..., J$ are increasing in the obvious way: from $0$ to $.1$. In particular, $F$ is $\#$5.
  41 KB (6,942 words) - 05:04, 22 June 2016
• Robustness of topology of digital images and point clouds by Saveliev
  Since increasing threshold $r$ enlarges the
  27 KB (4,547 words) - 04:08, 6 November 2012
• Connectivity
  ...nly by choosing more and more complex ways to compute the length (roughly, increasing the degree of the approximation of the curve). The choice of connectivity i
  4 KB (636 words) - 14:53, 9 October 2010
• Chain complexes
  ...ground up, using nothing but cells attached to each other in the order of increasing dimensions. In our box, we have: the parts, the glue, the schematics, and a
  33 KB (5,293 words) - 03:06, 31 March 2016
• Midtearm
  ...n below. Describe its behavior the function using words "decreasing" and "increasing".
  2 KB (308 words) - 15:23, 2 March 2016
• PageRank: testing
  ...ages are computed and displayed on the right. The scores for A, ..., J are increasing in the obvious way: from 0 to .1. In particular, F is #5.
  3 KB (546 words) - 16:08, 23 October 2011

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