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Chapter 1: Problem 49
Write a brief description of the meaning of the notation \(\lim _{x \rightarrow 8} f(x)=25\)
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After an object falls for \(t\) seconds, the speed \(S\) (in feet per second) of the object is recorded in the table. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline t & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline S & 0 & 48.2 & 53.5 & 55.2 & 55.9 & 56.2 & 56.3 \\ \hline \end{array} $$ (a) Create a line graph of the data. (b) Does there appear to be a limiting speed of the object? If there is a limiting speed, identify a possible cause.
Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ f(x)=x^{3}+3 x-3 $$
Let \(f(x)=\left(\sqrt{x+c^{2}}-c\right) / x, c>0 .\) What is the domain of \(f ?\) How can you define \(f\) at \(x=0\) in order for \(f\) to be continuous there?
Show that the Dirichlet function \(f(x)=\left\\{\begin{array}{ll}0, & \text { if } x \text { is rational } \\\ 1, & \text { if } x \text { is irrational }\end{array}\right.\) is not continuous at any real number.
Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arccos \frac{x}{4} $$
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