91Ó°ÊÓ

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty} x^{3}\left(\frac{1}{x}-\sin \frac{1}{x}\right)$$

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=-32 ; v(0)=20, s(0)=0$$

Use \(a\) graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity. $$f(x)=\frac{x \sin x}{x^{2}+1} \text { on }[-2 \pi, 2 \pi]$$

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(x^{2} e^{1 / x}-x^{2}-x\right)$$

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{aligned}&f^{\prime \prime}(x)>0 \text { on }(-\infty,-2) ; f^{\prime \prime}(-2)=0 ; f^{\prime}(-1)=f^{\prime}(1)=0\\\&f^{\prime \prime}(2)=0 ; f^{\prime}(3)=0 ; f^{\prime \prime}(x)>0 \text { on }(4, \infty)\end{aligned}$$

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=4 ; v(0)=-3, s(0)=2$$

Use \(a\) graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity. $$f(x)=x / \ln x$$

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{x-1}-\frac{1}{\sqrt{x-1}}\right)$$

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=0.2 t ; v(0)=0, s(0)=1$$

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{aligned}&f(-2)=f^{\prime \prime}(-1)=0 ; f^{\prime}\left(-\frac{3}{2}\right)=0 ; f(0)=f^{\prime}(0)=0\\\&f(1)=f^{\prime}(1)=0\end{aligned}$$