91Ó°ÊÓ

When a wholesaler sold a product at \(\$ 40\) per unit, sales were 300 units per week. After a price increase of \(\$ 5\), however, the average number of units sold dropped to 275 per week. Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=\left\\{\begin{array}{r}x^{2}+4, x<0 \\ 4-x, x \geq 0\end{array}\right.\)

Find an equation of the tangent line to the function at the given point. Then find the function values and the tangent line values at \(f(x+\Delta x)\) and \(y(x+\Delta x)\) for \(\Delta x=-0.01\) and \(0.01\). \(f(x)=3 x^{2}-1\) \((2,11)\)

Use a graphing utility or spreadsheet software program to complete the table. Then use the result to estimate the limit of \(f(x)\) as \(x\) approaches infinity. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ \(f(x)=\frac{2 x^{2}}{x+1}\)

A rectangular page is to contain 30 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page such that the least amount of paper is used.

Use a graphing utility or spreadsheet software program to complete the table. Then use the result to estimate the limit of \(f(x)\) as \(x\) approaches infinity. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ \(f(x)=\frac{x^{2}-1}{0.02 x^{2}}\)

Find an equation of the tangent line to the function at the given point. Then find the function values and the tangent line values at \(f(x+\Delta x)\) and \(y(x+\Delta x)\) for \(\Delta x=-0.01\) and \(0.01\). \(f(x)=\frac{x}{x^{2}+1}\) \((0,0)\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=\frac{x^{2}}{x^{2}+3}\)

A rectangle is bounded by the \(x\) - and \(y\) -axes and the graph of \(y=(6-x) / 2\) (see figure). What length and width should the rectangle have so that its area is a maximum?

A right triangle is formed in the first quadrant by the \(x\) - and \(y\) -axes and a line through the point \((1,2)\) (see figure). (a) Write the length \(L\) of the hypotenuse as a function of \(x\). (b) Use a graphing utility to approximate \(x\) graphically such that the length of the hypotenuse is a minimum. (c) Find the vertices of the triangle such that its area is a minimum.