By Sergiy Klymchuk
Counterexamples in Calculus serves as a supplementary source to reinforce the educational adventure in unmarried variable calculus classes. This publication positive aspects conscientiously developed wrong mathematical statements that require scholars to create counterexamples to disprove them. equipment of manufacturing those improper statements fluctuate. every now and then the communicate of a widely known theorem is gifted. In different cases the most important stipulations are passed over or altered or unsuitable definitions are hired. mistaken statements are grouped topically with sections dedicated to: features, Limits, Continuity, Differential Calculus and imperative Calculus.
This publication goals to fill a spot within the literature and supply a source for utilizing counterexamples as a pedagogical instrument within the learn of introductory calculus.
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Extra resources for Counterexamples in calculus
X/ is also discontinuous at x D a. x/ D sin2 x ; x2 if x ¤ 0 1; if x D 0 is continuous at the point x D 0. 4 A function always has a local maximum between any two local minima. Counterexample The functions yD x 4 C 0:1 x2 and y D sec2 x have no maximum between two local minima: ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2012/7/28 — 0:02 — page 49 — #59 ✐ ✐ 49 3. 5 For a continuous function there is always a strict local maximum between any two local minima. Counterexample The continuous function below does not have a strict local maximum between its two local minima.
Master” — 2012/7/28 — 0:02 — page 35 — #45 ✐ ✐ 35 1. x/ is unbounded and nonnegative for all real x, then it cannot have roots xn such that xn ! 1 as n ! 1. Counterexample The function y D jx sin xj has infinitely many roots xn such that xn ! 1 as n ! 1. x/ defined on Œa; b such that its graph does not contain any pieces of a horizontal straight line cannot take its extreme value infinitely many times on Œa; b. Counterexample The function ( sin x1 ; yD 0; if x ¤ 0 if x D 0 takes its absolute maximum value (D 1) and its absolute minimum value (D 1) infinitely many times on any closed interval containing zero.
X/ dx 1 converges, then Z also converges. x/ dx a converges. 1 The tangent to a curve at a point is the line that touches the curve at that point, but does not cross it there. Counterexample a) The x-axis is the tangent line to the curve y D x 3 , but it crosses the curve at the origin. y = x3 2 -6 -4 -2 0 0 2 4 6 -2 b) The three straight lines just touch and do not cross the curve below at the point, but none of them is the tangent line to the curve at that point. 2 The tangent line to a curve at a point cannot touch the curve at infinitely many other points.