91Ó°ÊÓ

Geometry A right triangle has a hypotenuse of \(\sqrt{18 \text { inches. }}\) (a) Draw and label a diagram that shows the base \(x\) and height \(y\) of the triangle. (b) Write a function \(A(x)\) that represents the area of the triangle. (c) The triangle has a maximum area when \(x=3\) inches. Use a graphing utility to complete the table and observe the behavior of the function as \(x\) approaches 3\. Use the table to find \(\lim _{x \rightarrow 3} A(x)\) (d) Use the graphing utility to graph the area function. Verify that the area is maximum when \(x=3\) inches.

Use the limit process to find the slope of the graph of the function at the specified point. Use a graphing utility to confirm your result. \(f(x)=10 x-2 x^{2}, \quad(3,12)\)

Use the limit process to find the slope of the graph of the function at the specified point. Use a graphing utility to confirm your result. \(g(x)=5-2 x, \quad(1,3)\)

Finding the Limit of a Summation, (a) rewrite the sum as a rational function \(S(n)\) (b) use \(S(n)\) to complete the table, and (c) find \(\lim _{n \rightarrow \infty} S(n)\) $$\sum_{i=1}^{n}\left[1-\left(\frac{i}{n}\right)^{2}\right]\left(\frac{1}{n}\right)$$

Find a formula for the slope of the graph of \(f\) at the point \((x, f(x)) .\) Then use it to find the slope at the two given points. \(f(x)=\sqrt{x-4}\) (a) \((5,1)\) (b) \((8,2)\)

Finding the Area of a Region,complete the table to show the approximate area of the region bounded by the graph of \(f\) and the \(x\) -axis over the specified interval using the indicated numbers \(n\) of rectangles of equal width. Then find the exact area as \(n \rightarrow \infty\). $$\begin{array}{ll}{\text { Function }} & {\text { Interval }} \\ {f(x)=2 x+5} & {[0,4]}\end{array}$$

Horizontal Asymptotes and Limits at Infinity In Exercises \(29 - 34 ,\) use a graphing utility to graph the function and verify that the horizontal asymptote corresponds to the limit at infinity. $$ y = \frac { 5 x } { 1 - x ^ { 2 } } $$

Graphical, Numerical, and Algebraic Analysis, (a) graphically approximate the limit (if it exists) by using a graphing utility to graph the function, (b) numerically approximate the limit (if it exists by using the table feature of the graphing utility to create a table, and (c) algebraically evaluate the limit (if it exists) by the appropriate technique(s). $$ \lim _{x \rightarrow 0^{-}} \frac{\sqrt{x+2}-\sqrt{2}}{x} $$

Average cost The cost function for a certain model of MP3 player is given by \(C = 73 x + 25,000 ,\) where \(C\) is in dollars and \(x\) is the number of MP3 players produced. (a) Write a model for the average cost per unit produced. (b) Find the average costs per unit when \(x = 1000\) and \(x = 5000\) . (c) Determine the limit of the average cost function as \(x\) approaches infinity. Explain the meaning of the limit in the context of the problem.

Evaluating a Limit from Calculus $$ \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $$ $$ f(x)=2 x+1 $$