91Ó°ÊÓ

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 \(f(x)0,\) where \(h=g-f .\)

Coughing forces the trachea (windpipe) to contract, which affects the velocity \(v\) of the air passing through the trachea. The velocity of the air during coughing is \(v=k(R-r) r^{2}, \quad 0 \leq r

The function \(s(t)\) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time \(t \geq 0\), (b) identify the time interval(s) when the particle is moving in a positive direction, (c) identify the time interval(s) when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction. $$ s(t)=t^{2}-7 t+10 $$

The function \(s(t)\) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time \(t \geq 0\), (b) identify the time interval(s) when the particle is moving in a positive direction, (c) identify the time interval(s) when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction. $$ s(t)=t^{3}-5 t^{2}+4 t $$

The function \(s(t)\) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time \(t \geq 0\), (b) identify the time interval(s) when the particle is moving in a positive direction, (c) identify the time interval(s) when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction. $$ s(t)=t^{3}-20 t^{2}+128 t-280 $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. An \(n\) th-degree polynomial has at most \((n-1)\) critical numbers.

Consider \(\lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of \(N\) that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\).