I have taught high school (certified K-12 educator), test prep, community college, and university level courses. I am passionate about education. My YouTube channel, with many videos I made during the pandemic: https://youtube.com/channel/UCESEJMP1XbubLjfR5seXgwQ
Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.
$ \displaystyle \lim_{x \to 0^-}f(x) = -1 $, $ \displaystyle \lim_{x \to 0^+}f(x) = 2 $, $ f(0) = 1 $
(a) What is wrong with the following equation?$$ \frac {x^2 + x - 6}{x - 2} = x + 3 $$
(b) In view of part (a), explain why the equation $$ \lim_{x \to 2}\frac {x^2 + x - 6}{x - 2} = \lim_{x \to 2}(x + 3) $$is correct.
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.
$ f(x) = (x + 2x^3)^4, \hspace{5mm} a = -1 $
Suppose $ f $ is continuous on $ [1, 5] $ and the only solutions of the equation $ f(x) = 6 $ are $ x = 1 $ and $ x = 4 $. If $ f(2) = 8$, explain why $ f(3) > 6 $.
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.
$ x^4 + x - 3 = 0 $, $ (1, 2) $
$ \ln x = x - \sqrt{x} $, $ (2, 3) $
Form a binary search tree for the words mathematics, physics, geography, zoology, meteorology, geology, psychology and chemistry (using alphabetical order).
