91Ó°ÊÓ

Determine whether the function is differentiable at \(x=2\). \(f(x)=\left\\{\begin{array}{ll}\frac{1}{2} x+1, & x<2 \\ \sqrt{2 x}, & x \geq 2\end{array}\right.\)

The stopping distance of an automobile, on dry, level pavement, traveling at a speed \(v\) (kilometers per hour) is the distance \(R\) (meters) the car travels during the reaction time of the driver plus the distance \(B\) (meters) the car travels after the brakes are applied (see figure). The table shows the results of an experiment. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Speed, } \boldsymbol{v} & 20 & 40 & 60 & 80 & 100 \\\\\hline \begin{array}{l}\text { Reaction Time } \\\\\text { Distance, } \boldsymbol{R}\end{array} & 8.3 & 16.7 & 25.0 & 33.3 & 41.7 \\\\\hline \begin{array}{l}\text { Braking Time } \\\\\text { Distance, } \boldsymbol{B}\end{array} & 2.3 & 9.0 & 20.2 & 35.8 & 55.9 \\\\\hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find a linear model for reaction time distance. (b) Use the regression capabilities of a graphing utility to find a quadratic model for braking distance. (c) Determine the polynomial giving the total stopping distance \(T\). (d) Use a graphing utility to graph the functions \(R, B\), and \(T\) in the same viewing window. (e) Find the derivative of \(T\) and the rates of change of the total stopping distance for \(v=40, v=80\), and \(v=100\) (f) Use the results of this exercise to draw conclusions about the total stopping distance as speed increases.

The displacement from equilibrium of an object in harmonic motion on the end of a spring is \(y=\frac{1}{3} \cos 12 t-\frac{1}{4} \sin 12 t\) where \(y\) is measured in feet and \(t\) is the time in seconds. Determine the position and velocity of the object when \(t=\pi / 8 .\)

Verify that the average velocity over the time interval \(\left[t_{0}-\Delta t, t_{0}+\Delta t\right]\) is the same as the instantaneous velocity at \(t=t_{0}\) for the position function \(s(t)=-\frac{1}{2} a t^{2}+c\)

The cost of producing \(x\) units of a product is \(C=60 x+1350\). For one week management determined the number of units produced at the end of \(t\) hours during an eight-hour shift. The average values of \(x\) for the week are shown in the table. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{t} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{x} & 0 & 16 & 60 & 130 & 205 & 271 & 336 & 384 & 392 \\ \hline \end{array} $$ (a) Use a graphing utility to fit a cubic model to the data. (b) Use the Chain Rule to find \(d C / d t\). (c) Explain why the cost function is not increasing at a constant rate during the 8 -hour shift.

Find an equation of the parabola \(y=a x^{2}+b x+c\) that passes through \((0,1)\) and is tangent to the line \(y=x-1\) at \((1,0)\).

Find a second-degree polynomial \(f(x)=a x^{2}+b x+c\) such that its graph has a tangent line with slope 10 at the point \((2,7)\) and an \(x\) -intercept at \((1,0)\).

Consider the third-degree polynomial \(f(x)=a x^{3}+b x^{2}+c x+d, \quad a \neq 0\) Determine conditions for \(a, b, c\), and \(d\) if the graph of \(f\) has (a) no horizontal tangents, (b) exactly one horizontal tangent, and (c) exactly two horizontal tangents. Give an example for each case.