By John H Mason, Sergiy Klymchuk
This booklet makes available to calculus scholars in highschool, university and collage a number of counter-examples to conjectures that many scholars erroneously make. additionally, it urges readers to build their very own examples by means of tinkering with those proven right here that allows you to improve the instance areas to which they've got entry, and to deepen their appreciation of conspectus and prerequisites using to theorems.
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Sample text
How could the conjecture be modified to make it correct? x) — (mx + c)) = 0. (b) A straight line is called a non-vertical asymptote to a curve as x tends to infinity if the curve gets closer and closer to the straight line (as close as we like) as x tends to infinity but does not touch or cross it. As x tends to infinity the function y = ^ ^ gets closer to the x-axis from above and below and 0) = 0. According to the first definition the x-axis is the non-vertical asymptote of lim> >(* the function y = ^ , but its graph crosses the x-axis infinitely many times, so the definitions (a) and (b) are not equivalent.
Fix) = if x ^ a , x—a 1 , if x ^ a (x) =x+ g x—a f{x) = g(x) = - , ifx = a. Both functions f{x) and g{x) are discontinuous at x = a but the function JC, if x ^ a /(*) + *(*) = a, if x = a is continuous at x = a. For example, if a = 2: Let h{x) be any continuous function on an interval (a,'b)9 and d{x) a function discontinuous at x = c in that interval. Then /i(jc) - d{x) is discontinuous at x = c as is /i(x) + d(x)9 and their sum is continuous at x = c. How might this construction be generalised?
20 If a function y = g(x) is not differentiable at x = a and a function y = /(JC) is not differentiable at g(a) then the function F(x) = f(g(x)) is not differentiable at x = a. 21 If a function y — f(x) is defined on [a, b]> differentiable on (a, b) and f(a) = f(b), then there exists a point c e (a, b) such that f(c) = 0. 22 If a function is twice-differentiable in a certain neighbourhood of the point x = a and its second derivative is zero at that point then the point (a, f(a)) is a point of inflection for the graph of the function.