91Ó°ÊÓ

\(\displaystyle\sum_{i=1}^{n} i^3 = \) _______________

The limit \(\lim_{x \to c^{-}} f(x)=L_1\) is an example of a _______ _______ .

The exact _______ of a plane region \(R\) is given by the limit of the sum of \(n\) rectangles as \(n\) approaches \(\infty\).

The slope of the tangent line to a graph at \((x, f(x))\) is given by _______ .

To evaluate the limit of a polynomial function, use _______ _______.

A sequence that does not have a limit is said to ________.

GEOMETRY You create an open box from a square piece of material 24 centimeters on a side. You cut equal squares from the corners and turn up the sides. (a) Draw and label a diagram that represents the box. (b) Verify that the volume \(V\) of the box is given by \(V=4x(12-x)^2\). (C) The box has a maximum volume when \(x=4\). Use a graphing utility to complete the table and observe the behavior of the function as \(x\) approaches 4. Use the table to find \(\lim_{x \to 4} V\). (d) Use a graphing utility to graph the volume function. Verify that the volume is maximum when \(x=4\).

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{60} 7$$

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{45} 3$$

GEOMETRY You are given wire and are asked to forma right triangle with a hypotenuse of \(\sqrt{18}\) inches whose area is as large as possible. (a) Draw and label a diagram that shows the base \(x\) and height \(y\) of the triangle. (b) Verify that the area \(A\) of the triangle is given by \(A=\frac{1}{2}x \sqrt{18-x^{2}}\). (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 \(\lim_{x \to 3} A\). (d) Use a graphing utility to graph the area function.Verify that the area is maximum when \(x=3\) inches.