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Chapter 8: Problem 23
Find the indefinite integral. $$ \int \frac{\sec x \tan x}{\sec x-1} d x $$
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Inventory The stockpile level of liquefied petroleum gases in the United States in 2006 can be approximated by the model \(Q=109+32 \cos \frac{\pi(t+3)}{6}\) where \(Q\) is measured in millions of barrels and \(t\) is the time in months, with \(t=1\) corresponding to January. Find the average levels given by this model during $$ \begin{array}{l}{\text { (a) the first quarter }(0 \leq t \leq 3)} \\ {\text { (b) the second quarter }(3 \leq t \leq 6)} \\ {\text { (c) the entire year }(0 \leq t \leq 12)}\end{array} $$
Health For a person at rest, the velocity \(v\) (in liters per second) of air flow into and out of the lungs during a respiratory cycle is given by \(v=0.9 \sin \frac{\pi t}{3}\) where \(t\) is the time in seconds. Inhalation occurs when \(v>0,\) and exhalation occurs when \(v<0 .\) (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Use a graphing utility to graph the velocity function.
sketch the graph of the function by hand. Use a graphing utility to verify your sketch. $$ y=-2 \sin 6 x $$
complete the table (using a spreadsheet or a graphing utility set in radian mode) to estimate \(\lim _{x \rightarrow 0} f(x)\). $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{\tan 2 x}{x} $$
find the period of the function. $$ y=3 \sec 5 x $$
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