91Ó°ÊÓ

Minimum Distance Sketch the graph of \(f(x)=2-2 \sin x\) on the interval \([0, \pi / 2]\) (a) Find the distance from the origin to the \(y\) -intercept and the distance from the origin to the \(x\) -intercept. (b) Write the distance \(d\) from the origin to a point on the graph of \(f\) as a function of \(x\) . Use your graphing utility to graph \(d\) and find the minimum distance. (c) Use calculus and the zero or root feature of a graphing utility to find the value of \(x\) that minimizes the function \(d\) on the interval \([0, \pi / 2] .\) What is the minimum distance?

Describing Terms When using differentials, what is meant by the terms propagated error, relative error, and percent error?

Finding Absolute Extrema In Exercises \(41-44,\) use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. $$ f(x)=\sqrt{x}+\cos \frac{x}{2}, \quad[0,2 \pi] $$

Use a graphing utility to (a) graph the function \(f\) on the given interval, (b) find and graph the secant line through points on the graph of \(f\) at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of \(f\) that are parallel to the secant line. \(f(x)=\frac{x}{x+1}, \quad\left[-\frac{1}{2}, 2\right]\)

Comparing Functions In Exercises 55 and \(56,\) use symmetry, extrema, and zeros to sketch the graph of \(f .\) How do the functions \(f\) and \(g\) differ? $$ \begin{array}{l}{f(x)=\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1}} \\\ {g(x)=x\left(x^{2}-3\right)}\end{array} $$

Honeycomb The surface area of a cell in a honeycomb is $$S=6 h s+\frac{3 s^{2}}{2}\left(\frac{\sqrt{3}-\cos \theta}{\sin \theta}\right)$$ where \(h\) and \(s\) are positive constants and \(\theta\) is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle \(\theta(\pi / 6 \leq \theta \leq \pi / 2)\) that minimizes the surface area \(S\) .

Finding a Cubic Function In Exercises 61 and \(62,\) find \(a,\) \(b, c,\) and \(d\) such that the cubic $$f(x)=a x^{3}+b x^{2}+c x+d$$ satisfies the given conditions. Relative maximum: \((3,3)\) Relative minimum: \((5,1)\) Inflection point: \((4,2)\)

Consider the function $$ f(x)=\frac{2 x^{n}}{x^{4}+1} $$ for nonnegative integer values of \(n .\) (a) Discuss the relationship between the value of \(n\) and the symmetry of the graph. (b) For which values of \(n\) will the \(x\) -axis be the horizontal asymptote? (c) For which value of \(n\) will \(y=2\) be the horizontal asymptote? (d) What is the asymptote of the graph when \(n=5 ?\) (e) Use a graphing utility to graph \(f\) for the indicated values of \(n\) in the table. Use the graph to determine the number of extrema \(M\) and the number of inflection points \(N\) of the graph.

Average Cost A manufacturer has determined that the total cost \(C\) of operating a factory is $$C=0.5 x^{2}+15 x+5000$$ where \(x\) is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is \(C / x . )\)

Point of Inflection and Extrema Show that the point of inflection of $$f(x)=x(x-6)^{2}$$ lies midway between the relative extrema of \(f\)