Chapter 10

Find the linear approximation of \(f(x)=(1+x)^{1 / 4}\) at \(x=0\) and use the equation to approximate \(f(0.1)\).

Find the approximate value of \(\sqrt[3]{28}\) using linear approximation.

Find the approximate value of \(\cos 46^{\circ}\) using linear approximation.

Find the point on the graph of \(y=|x|^{3}\) such that the tangent at the point is parallel to the line \(y-12 x=3\).

If the position function of a particle is \(s(t)=\frac{t^{3}}{3}-3 t^{2}+4\) find the velocity and position of the particle when its acceleration is 0 .

A ball is dropped from the top of a 640 -foot building. The position function of the ball is \(s(t)=-16 t^{2}+640,\) where \(t\) is measured in seconds and \(s(t)\) is in feet. Find: (a) The position of the ball after 4 seconds. (b) The instantaneous velocity of the ball at \(t=4\). (c) The average velocity for the first 4 seconds. (d) When the ball will hit the ground. (e) The speed of the ball when it hits the ground.

The position function of a particle moving on a line is \(s(t)=t^{3}-3 t^{2}+\) 1, \(t \geq 0\) where \(t\) is measured in seconds and \(s\) in meters. Describe the motion of the particle.

Find the linear approximation of \(f(x)=\sin x\) at \(x=\pi\). Use the equation to find the approximate value of \(f\left(\frac{181 \pi}{180}\right)\).

Find the linear approximation of \(f(x)=\ln (1+x)\) at \(x=2\).

Find the value(s) of \(x\) at which the graphs of \(y=\ln x\) and \(y=x^{2}+3\) have parallel tangents.

The position functions of two moving particles are \(s 1(t)=\ln t\) and \(s\) \(2(t)=\sin t\) and the domain of both functions is \(1 \leq t \leq 8 .\) Find the values of \(t\) such that the velocities of the two particles are the same.

The position function of a moving particle on a line is \(s(t)=\sin (t)\) for \(0 \leq t \leq 2 \pi .\) Describe the motion of the particle.

A coin is dropped from the top of a tower and hits the ground 10.2 seconds later. The position function is given as \(s(t)=-16 t^{2}-v_{0} t+s_{0}\), where \(s\) is measured in feet, \(t\) in seconds, and \(v_{0}\) is the initial velocity and \(s_{0}\) is the initial position. Find the approximate height of the building to the nearest foot.

Given \(f(x)=x^{3}-3 x^{2}+3 x-1\) and the point (1,2) is on the graph of \(f\) \({ }^{1}(x)\). Find the slope of the tangent line to the graph of \(f^{1}(x)\) at (1,2)

Evaluate \(\lim _{x \rightarrow 100} \frac{x-100}{\sqrt{x}-10}\)

A function \(f\) is continuous on the interval (-1,8) with \(f(0)=0, f(2)=\) \(3,\) and \(f(8)=1 / 2\) and has the following properties: $$\begin{array}{c|c|c|c|c|c}\hline \text { INTERVALS } & (-1,2) & \mathbf{x}=\mathbf{2} & \mathbf{( 2 , 5 )} & \mathbf{x}=\mathbf{5} &\mathbf{( 5 , 8 )} \\ \hline f^{\prime} & \+ & 0 & \- & \- & \- \\\\\hline f^{\prime \prime} & \- & \- & \- & 0 & \+ \\\\\hline\end{array}$$ (a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find where \(f\) has its absolute extrema. (c) Find where \(f\) has the points of inflection. (d) Find the intervals on which \(f\) is concave upward or downward. (e) Sketch a possible graph of \(f\).