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• 1983 AIME Problems/Problem 9
  ...value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>. Since <math>x>0</math>, and <math>\sin{x}>0</math> because <math>0< x<\pi</math>, we have <math>y>0</math>. So we can apply [[AM-GM]]:
  4 KB (722 words) - 20:25, 14 January 2023
• Holomorphic function
  to <math>x</math> and <math>y</math> are continuous on <math>D</math>, and the following system holds at <math>z</math>; thus the derivative of <math>f</math> is continuous at <math>z</math>, and so
  9 KB (1,537 words) - 21:04, 26 July 2017
• 2006 Romanian NMO Problems
  ...ngle with the base <math>BC</math>. We know that <math>\angle ABD = \frac{\pi}{2}</math>. Let <math>M</math> be the midpoint of <math>BC</math>. The poin Prove that for all <math>\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)</math> and <math>\displaystyle n \in \mathbb N^\ast</math> we h
  11 KB (1,779 words) - 14:57, 7 May 2012
• Tangent (geometry)
  ...math> intersects it in 2 points and the tangent line at <math>\left(\frac{\pi}2, 1\right)</math> intersects it in [[infinite]]ly many points (and is in f ...- only [[continuous]] functions may have tangent lines, and there are many continuous functions which fail to have tangent lines either at some points (for insta
  2 KB (332 words) - 21:54, 11 March 2024
• Argument
  ...pi, \pi]</math>. This forces us to state this equality [[modulo]] <math>2\pi</math>.
  1 KB (188 words) - 20:59, 31 July 2020
• L'Hôpital's Rule
  Just so all of us know <math>\eta(h)</math> is a function that is both continuous and has a limit of <math>0</math> as the <math>h</math> in the derivative f (For example, <math>g(x) = \cos\left(\frac{\pi}{2} x\right)</math>, <math>f(x) = 1-x</math>, and <math>a = 1</math>.)
  2 KB (475 words) - 15:04, 24 March 2022
• 2009 AMC 12B Problems
  ...and independently for each face. What is the probability that there is a continuous stripe encircling the cube? For how many values of <math>x</math> in <math>[0,\pi]</math> is <math>\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)</math>?
  13 KB (2,030 words) - 03:04, 5 September 2021
• Cauchy's Integral Formula
  <cmath> \frac{1}{2\pi i} \int\limits_C \frac{f(z)}{z- z_0}dz = f(z_0) .</cmath> as <math>h(t) = r e^{it}+ z_0</math>, for <math>t\in [0,2\pi]</math>. Since <math>\int\limits_{C_r}
  4 KB (689 words) - 17:19, 18 January 2024
• Fundamental group
  ...motopy]] between <math>f</math> and <math>g</math> (that is, if there is a continuous function <math>F:[0,1]\times[0,1]\to X</math> with <math>F(a,0)=f(a)</math> ...ontinuous map <math>f:(X,x_0)\to(Y,y_0)</math> (that is, <math>f</math> is continuous and <math>f(x_0) = y_0</math>), one may define a group homomorphism <math>f
  8 KB (1,518 words) - 20:11, 23 January 2017
• 1952 AHSME Problems
  \text{(D) the area of the circle is } \pi \text{ times the area of the square}\qquad\\ To be continuous at <math>x = - 1</math>, the value of <math>\frac {x^3 + 1}{x^2 - 1}</math>
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1972 IMO Problems/Problem 6
  ...asing order of their <math>z</math>-coordinates. First take a plane <math>\pi</math> orthogonal to <math>\pi_i</math>, which cuts <math>\pi_1,\pi_2,\pi_3 ..._1,P_2</math>, these will be continuous functions of the angle that <math>\pi</math> makes with <math>\pi_i</math>, and for one of the points <math>P_1,P
  7 KB (1,370 words) - 15:42, 29 January 2021
• 2013 Mock AIME I Problems/Problem 6
  ...te that a minimum of <math>-\sqrt{74}</math> can be attained at <math>f(x+\pi)</math>. Thus the values of <math>k</math> that work are the integers from
  732 bytes (126 words) - 09:22, 23 October 2023
• Normal distribution
  The normal distribution is a [[continuous]] probability distribution that is widely used in [[statistics]] and beyond ...f a <math>\mathcal{N}(\mu, \sigma)</math> is <math>f(x) = \frac{1}{\sqrt{2\pi\sigma}} e^{-\frac{(x-\mu)^2}{2\sigma}}</math>, and the [[cdf|cumulative den
  1 KB (214 words) - 22:32, 12 April 2020
• 2020 INMO Problems/Problem 2
  <math>\emph{Proof.}</math> Note that <math>\theta=\frac{\pi}{2} \implies P(1)=P(-1)</math>. Set <math>c=P(1)</math>. Then <math>(1-x^2) ...math>\mathbb{R}</math>, the equation extends to these as <math>Q</math> is continuous.)
  2 KB (397 words) - 13:02, 5 November 2020
• Polya’s method for extremums
  Note that the shortest (continuous) curve has ends <math>D</math> and <math>E</math> on two sides of the trian ...he blue line will coincide, that is, the set of these two curves is also a continuous curve.
  12 KB (2,104 words) - 14:11, 24 February 2024