Chapter 12

\(\displaystyle\sum_{i=1}^{n} c = \) _______________, \(c\) is a constant.

_______ is the study of the rates of change of functions.

To evaluate the limit of a rational function that has common factors in its numerator and denominator,use the _______ _______ _______ .

If \(f(x)\) becomes arbitrarily close to a unique number \(L\) as \(x\) approaches \(c\) from either side, the _______ of \(f(x)\) as \(x\) approach \(c\) is \(L\).

When evaluating limits at infinity for complicated rational functions, you can divide the numerator and denominator by the ________ term in the denominator.

The _______ _______ to the graph of a function at a point is the line that best approximates the slope of the graph at the point.

The fraction \(\frac{0}{0}\) has no meaning as a real number and therefore is called an _______ _______ .

An alternative notation for \(\lim_{x\to c}f(x) = L\) is \(f(x) \rightarrow L \) as \(x \rightarrow c \), which is read as "\(f(x)\) _______ \(L\) as \(x\) _______ \(c\)".

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

A sequence that has a limit is said to ________.

A _______ _______ is a line through the point of tangency and a second point on the graph.

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 \(\infty\).

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

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 _______ _______.

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

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. $$\displaystyle\sum_{i=1}^{45} 3$$

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.

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{20} i^3$$

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 2}\ (5x+4)$$

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{30} i^2$$

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 1}\ (2x^2+x-4)$$

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{k=1}^{20} (k^3 + 2)$$

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \left(\dfrac{3}{x^2} + 1 \right) \\]

In Exercises 9-16, 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. \(g(x) = x^2-4x, \quad (3, -3)\)

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 6} \dfrac{x-6}{x^2-36}$$

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 3}\ \dfrac{x-3}{x^2 -9}$$

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{k=1}^{50} (2k + 1)$$

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \left(\dfrac{4}{3x} - 5 \right) \\]

In Exercises 9-16, 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. \(f(x) = 10x-2x^2, \quad (3, 12)\)

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 7} \dfrac{7-x}{x^2-49}$$

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to -1}\ \dfrac{x+1}{x^2 -x-2}$$

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{j=1}^{25} (j^2 + j)$$

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \left(\dfrac{1-x}{1+x} \right) \\]

In Exercises 9-16, 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. \(g(x) = 5-2x, \quad (1, 3)\)

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to -1} \dfrac{1-2x-3x^2}{1+x}$$

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 0}\ \dfrac{\sin\ 2x}{x}$$

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{j=1}^{10} (j^3 - 3j^2)$$

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \left(\dfrac{1+5x}{1-4x} \right) \\]

In Exercises 9-16, 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. \(h(x) = 2x+5, \quad (-1, 3)\)

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to -2} \dfrac{x^2+6x+8}{x+2}$$

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 0}\ \dfrac{\tan\ x}{2x}$$

In Exercises 13-20, (a) rewrite the sum as a rational function \(S(n)\), (b) use \(S(n)\) to complete the table, and (c) find \(\lim_{n \to \infty} S(n)\). $$\displaystyle\sum_{i=1}^{n} \dfrac{i^3}{n^4}$$

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \dfrac{4x-3}{2x+1} \\]

In Exercises 9-16, 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. \(g(x) = \dfrac{4}{x}, \quad (2, 2)\)

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to -1} \dfrac{1-2x-3x^2}{1+x}$$

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 1} \dfrac{x-1}{x^2+2x-3}$$

In Exercises 13-20, (a) rewrite the sum as a rational function \(S(n)\), (b) use \(S(n)\) to complete the table, and (c) find \(\lim_{n \to \infty} S(n)\). $$\displaystyle\sum_{i=1}^{n} \dfrac{i}{n^2}$$