E-coli Growth A strain of E-coli Beu 397-recA441 is placed into a nutrient broth at 30

The interval can be written as the inequality

The slope of the line containing the points and is

Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin.

Write the pointslope form of the line with slope 5 containing the point

The intercepts of the equation are

A function is on an open interval I if, for any choice of and in I, with we have

A(n) function is one for which for every x in the domain of a(n) function is one for which for every x in the domain of

True or False A function is decreasing on an open interval I if, for any choice of and in I, with we have f1x12 7 f1x22

True or False A function has a local maximum at c if there is an open interval I containing c so that for all in I,

True or False Even functions have graphs that are symmetric with respect to the origin.

In Problems 1120, use the graph of the function given. Is increasing on the interval

In Problems 1120, use the graph of the function given. Is decreasing on the interval

In Problems 1120, use the graph of the function given. Is increasing on the interval

In Problems 1120, use the graph of the function given. Is decreasing on the interval

In Problems 1120, use the graph of the function given. List the interval(s) on which is increasing.

In Problems 1120, use the graph of the function given. List the interval(s) on which is decreasing.

In Problems 1120, use the graph of the function given. Is there a local maximum value at 2? If yes, what is it?

In Problems 1120, use the graph of the function given. Is there a local maximum value at 5? If yes, what is it?

In Problems 1120, use the graph of the function given. List the number(s) at which has a local maximum. What are the local maximum values?

In Problems 1120, use the graph of the function given. List the number(s) at which has a local minimum. What are the local minimum values?

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2128, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant

In Problems 2932, the graph of a function is given. Use the graph to find: (a) The numbers, if any, at which has a local maximum value. What are the local maximum values? (b) The numbers, if any, at which has a local minimum value. What are the local minimum values?

In Problems 2932, the graph of a function is given. Use the graph to find: (a) The numbers, if any, at which has a local maximum value. What are the local maximum values? (b) The numbers, if any, at which has a local minimum value. What are the local minimum values?

In Problems 2932, the graph of a function is given. Use the graph to find: (a) The numbers, if any, at which has a local maximum value. What are the local maximum values? (b) The numbers, if any, at which has a local minimum value. What are the local minimum values?

In Problems 2932, the graph of a function is given. Use the graph to find: (a) The numbers, if any, at which has a local maximum value. What are the local maximum values? (b) The numbers, if any, at which has a local minimum value. What are the local minimum values?

In Problems 3344, determine algebraically whether each function is even, odd, or neither. f1x2 = 4x3

In Problems 3344, determine algebraically whether each function is even, odd, or neither. f1x2 = 2x - 5 4 - x2

In Problems 3344, determine algebraically whether each function is even, odd, or neither. g1x2 = -3x + 5 2 f1x2 = 2x - 5

In Problems 3344, determine algebraically whether each function is even, odd, or neither. h1x2 = 3x3 g1x2 = -3x + 5

In Problems 3344, determine algebraically whether each function is even, odd, or neither. F1x2 = 1

In Problems 3344, determine algebraically whether each function is even, odd, or neither. G1x2 = 1x

In Problems 3344, determine algebraically whether each function is even, odd, or neither. f1x2 = x + x

In Problems 3344, determine algebraically whether each function is even, odd, or neither. f1x2 = 23 2x2

In Problems 3344, determine algebraically whether each function is even, odd, or neither. g1x2 = 1 x2

In Problems 3344, determine algebraically whether each function is even, odd, or neither. h1x2 = x x2 - 1

In Problems 3344, determine algebraically whether each function is even, odd, or neither. h1x2 = -x3 3x2 - 9

In Problems 3344, determine algebraically whether each function is even, odd, or neither. F1x2 = 2x x

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 4552, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = x + 5 1-1, 32 3 - 3x + 2 1-2, 22

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = x3 - 3x2 f1x2 = x + 5 1-1, 3

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = x5 - x3 1-2, 22

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = x4 - x2 1-2, 22

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = -0.2x3 - 0.6x2 + 4x - 6 1-6, 42

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = -0.4x3 + 0.6x2 + 3x - 2 1-4, 52

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = 0.25x4 + 0.3x3 - 0.9x2 + 3 1-3, 22

In Problems 5360, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. f1x2 = -0.4x4 - 0.5x3 + 0.8x2 - 2 1-3, 22

Find the average rate of change of (a) From 0 to 2 (b) From 1 to 3 (c) From 1 to 4

Find the average rate of change of (a) From 0 to 2 (b) From 1 to 3 (c) From to 1

Find the average rate of change of (a) From to (b) From to 1 (c) From 1 to 3

Find the average rate of change of (a) From to 1 (b) From 0 to 2 (c) From 2 to 5

(a) Find the average rate of change from 1 to 3. (b) Find an equation of the secant line containing and

(a) Find the average rate of change from 2 to 5. (b) Find an equation of the secant line containing and

(a) Find the average rate of change from to 1. (b) Find an equation of the secant line containing and

(a) Find the average rate of change from to 2. (b) Find an equation of the secant line containing and

(a) Find the average rate of change from 2 to 4. (b) Find an equation of the secant line containing and

(a) Find the average rate of change from 0 to 3. (b) Find an equation of the secant line containing and 13, h1322.

(a) Determine whether g is even, odd, or neither. (b) There is a local minimum value of at 3. Determine the local maximum value.

(a) Determine whether f is even, odd, or neither. (b) There is a local maximum value of 16 at 2. Determine the local minimum value.

(a) Determine whether F is even, odd, or neither. (b) There is a local maximum value of 24 at . Determine a second local maximum value. (c) Suppose the area under the graph of F between and that is bounded below by the x-axis is 47.4 square units. Using the result from part (a), determine the area under the graph of F between and bounded below by the x-axis.

(a) Determine whether G is even, odd, or neither. (b) There is a local maximum value of 400 at Determine a second local maximum value. (c) Suppose the area under the graph of G between and that is bounded below by the x-axis is 1612.8 square units. Using the result from part (a), determine the area under the graph of G between x = -6 and bounded below by the x = 0 x-axis

Minimum Average Cost The average cost per hour in dollars, , of producing x riding lawn mowers can be modeled by the function (a) Use a graphing utility to graph . (b) Determine the number of riding lawn mowers to produce in order to minimize average cost. (c) What is the minimum average cost?

Medicine Concentration The concentration C of a medication in the bloodstream t hours after being administered is modeled by the function (a) After how many hours will the concentration be highest? (b) A woman nursing a child must wait until the concentration is below 0.5 before she can feed him. After taking the medication, how long must she wait before feeding her child?

E-coli Growth A strain of E-coli Beu 397-recA441 is placed into a nutrient broth at 30 Celsius and allowed to grow. The data shown in the table are collected. The population is measured in grams and the time in hours. Since population P depends on time t and each input corresponds to exactly one output, we can say that population is a function of time; so represents the population at time t. (a) Find the average rate of change of the population from 0 to 2.5 hours. (b) Find the average rate of change of the population from 4.5 to 6 hours. (c) What is happening to the average rate of change as time passes?

e-Filing Tax Returns The Internal Revenue Service Restructuring and Reform Act (RRA) was signed into law by President Bill Clinton in 1998. A major objective of the RRA was to promote electronic filing of tax returns. The data in the table that follows, show the percentage of individual income tax returns filed electronically for filing years 20002008. Since the percentage P of returns filed electronically depends on the filing year y and each input corresponds to exactly one output, the percentage of returns filed electronically is a function of the filing year; so represents the percentage of returns filed electronically for filing year y. (a) Find the average rate of change of the percentage of e-filed returns from 2000 to 2002. (b) Find the average rate of change of the percentage of e-filed returns from 2004 to 2006. (c) Find the average rate of change of the percentage of e-filed returns from 2006 to 2008. (d) What is happening to the average rate of change as time passes?

For the function compute each average rate of change: (a) From 0 to 1 (b) From 0 to 0.5 (c) From 0 to 0.1 (d) From 0 to 0.01 (e) From 0 to 0.001 (f) Use a graphing utility to graph each of the secant lines along with . (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?

For the function compute each average rate of change: (a) From 1 to 2 (b) From 1 to 1.5 (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with . (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = 2x + 5

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = -3x + 2

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = x

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = 2x

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = 2x + 3x - 2 2 - 3x + 1

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = -x2

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = 1 x

Problems 8188 require the following discussion of a secant line. The slope of the secant line containing the two points and on the graph of a function may be given as In calculus, this expression is called the difference quotient of (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find for 0.1, and 0.01 at What value does approach as h approaches 0? (c) Find the equation for the secant line at with (d) Use a graphing utility to graph and the secant line found in part (c) on the same viewing window. f1x2 = 1 x2

Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: and a local maximum value of is at a local minimum value of is at 2. Compare your graph with those of others. Comment on any differences.

Redo Problem 89 with the following additional information: increasing on decreasing on Again compare your graph with others and comment on any differences.

How many x-intercepts can a function defined on an interval have if it is increasing on that interval? Explain.

Suppose that a friend of yours does not understand the idea of increasing and decreasing functions. Provide an explanation, complete with graphs, that clarifies the idea.

Can a function be both even and odd? Explain.

Using a graphing utility, graph on the interval Use MAXIMUM to find the local maximum values on Comment on the result provided by the calculator.

A function f has a positive average rate of change on the interval 2, 5 . Is f increasing on 2, 5 ? Explain.

Show that a constant function has an average rate of change of 0. Compute the average rate of change of on the interval Explain how this can happen.