Chapter 3: Problem 8
Identify the open intervals on which the function is increasing or decreasing.
$$
h(x)=\cos \frac{x}{2}, 0
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Numerical, Graphical, and Analytic Analysis Consider the functions \(f(x)=x\)
and \(g(x)=\tan x\) on the interval \((0, \pi / 2)\)
(a) Complete the table and make a conjecture about which is the greater
function on the interval \((0, \pi / 2)\).
$$
\begin{array}{|l|l|l|l|l|l|l|}
\hline x & 0.25 & 0.5 & 0.75 & 1 & 1.25 & 1.5 \\
\hline f(x) & & & & & & \\
\hline g(x) & & & & & & \\
\hline
\end{array}
$$
(b) Use a graphing utility to graph the functions and use the graphs to make a
conjecture about which is the greater function on the interval \((0, \pi / 2)\).
(c) Prove that \(f(x)
In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=5-\frac{1}{x^{2}} $$
Show that the point of inflection of \(f(x)=x(x-6)^{2}\) lies midway between the relative extrema of \(f\).
Learning Theory In a group project in learning theory, a mathematical model for the proportion \(P\) of correct responses after \(n\) trials was found to be $$ P=\frac{0.83}{1+e^{-0.2 n}} $$ (a) Find the limiting proportion of correct responses as \(n\) approaches infinity. (b) Find the rates at which \(P\) is changing after \(n=3\) trials and \(n=10\) trials.
In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x^{3}}{\sqrt{x^{2}-4}} $$
What do you think about this solution?
We value your feedback to improve our textbook solutions.
