One of the fundamental themes of calculus is to find the slope of the tangent line to a

In Exercises 14, find the intercepts (if any). x32

In Exercises 14, find the intercepts (if any). x32y

In Exercises 14, find the intercepts (if any). 32y

In Exercises 14, find the intercepts (if any). 2y2

In Exercises 5 and 6, check for symmetry with respect to both axes and to the origin. 2y2x

In Exercises 5 and 6, check for symmetry with respect to both axes and to the origin. y2x2

In Exercises 714, sketch the graph of the equation. 2x2

In Exercises 714, sketch the graph of the equation. 2x2x

In Exercises 714, sketch the graph of the equation. x2xy

In Exercises 714, sketch the graph of the equation. 2xy

In Exercises 714, sketch the graph of the equation. xy

In Exercises 714, sketch the graph of the equation. xyx

In Exercises 714, sketch the graph of the equation. yx

In Exercises 714, sketch the graph of the equation. 1y1x

In Exercises 15 and 16, describe the viewing window of a graphing utility that yields the figure. y1xx

In Exercises 15 and 16, describe the viewing window of a graphing utility that yields the figure. 1xx

In Exercises 17 and 18, use a graphing utility to find the point(s) of intersection of the graphs of the equations. 1xxy

In Exercises 17 and 18, use a graphing utility to find the point(s) of intersection of the graphs of the equations. xxy2

Think About It Write an equation whose graph has intercepts at and and is symmetric with respect to the origin xy2

Think About It For what value of does the graph of pass through the point? (a) (b) ( y24

In Exercises 21 and 22, plot the points and find the slope of the line passing through the points. y24x

In Exercises 21 and 22, plot the points and find the slope of the line passing through the points. 24x

In Exercises 23 and 24, use the concept of slope to find t such that the three points are collinear 4xy

In Exercises 23 and 24, use the concept of slope to find t such that the three points are collinear 4xy3

In Exercises 2528, find an equation of the line that passes through the point with the indicated slope. Sketch the line. xy3y

In Exercises 2528, find an equation of the line that passes through the point with the indicated slope. Sketch the line. y3y

In Exercises 2528, find an equation of the line that passes through the point with the indicated slope. Sketch the line. y3y

In Exercises 2528, find an equation of the line that passes through the point with the indicated slope. Sketch the line. 3y9

Find equations of the lines passing through and having the following characteristics. (a) Slope of (b) Parallel to the line (c) Passing through the origin (d) Parallel to the -axis y9

Find equations of the lines passing through and having the following characteristics. (a) Slope of (b) Perpendicular to the line (c) Passing through the point (d) Parallel to the -axis 9x

Rate of Change The purchase price of a new machine is $12,500, and its value will decrease by $850 per year. Use this information to write a linear equation that gives the value of the machine years after it is purchased. Find its value at the end of 3 years. 9x2

Break-Even Analysis A contractor purchases a piece of equipment for $36,500 that costs an average of $9.25 per hour for fuel and maintenance. The equipment operator is paid $13.50 per hour, and customers are charged $30 per hour. (a) Write an equation for the cost of operating this equipment for hours. (b) Write an equation for the revenue derived from hours of use. (c) Find the break-even point for this equipment by finding the time at which 9x2y

In Exercises 3336, sketch the graph of the equation and use the Vertical Line Test to determine whether the equation expresses as a function of x2y

In Exercises 3336, sketch the graph of the equation and use the Vertical Line Test to determine whether the equation expresses as a function of x2yx

In Exercises 3336, sketch the graph of the equation and use the Vertical Line Test to determine whether the equation expresses as a function of 2yx

In Exercises 3336, sketch the graph of the equation and use the Vertical Line Test to determine whether the equation expresses as a function of yxx

Evaluate (if possible) the function at the specified values of the independent variable, and simplify the results. xx

Evaluate (if possible) the function at each value of the independent variable. xx2

Find the domain and range of each function. x2y

Given and evaluate each expression. x2y

Sketch (on the same set of coordinate axes) graphs of for and 2yx

Use a graphing utility to graph Use the graph to write a formula for the function shown in the figure. To print an enlarged copy of the graph, select the MathGraph button. 2yx3

Conjecture (a) Use a graphing utility to graph the functions and in the same viewing window. Write a description of any similarities and differences you observe among the graphs. Odd powers: Even powers: (b) Use the result in part (a) to make a conjecture about the graphs of the functions and Use a graphing utility to verify your conjecture. yx3

Think About It Use the result of Exercise 43 to guess the shapes of the graphs of the functions and Then use a graphing utility to graph each function and compare the result with your guess. x34

Area A wire 24 inches long is to be cut into four pieces to form a rectangle whose shortest side has a length of (a) Write the area of the rectangle as a function of (b) Determine the domain of the function and use a graphing utility to graph the function over that domain. (c) Use the graph of the function to approximate the maximum area of the rectangle. Make a conjecture about the dimensions that yield a maximum area x34x

Writing The following graphs give the profits for two small companies over a period of 2 years. Create a story to describe the behavior of each profit function for some hypothetical product the company produces. 34xy

Think About It What is the minimum degree of the polynomial function whose graph approximates the given graph? What sign must the leading coefficient have? 4xy

Stress Test A machine part was tested by bending it centimeters 10 times per minute until the time (in hours) of failure. The results are recorded in the table. (a) Use the regression capabilities of a graphing utility to find a linear model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the graph to determine whether there may have been an error made in conducting one of the tests or in recording the results. If so, eliminate the erroneous point and find the model for the remaining data. 1xy

Harmonic Motion The motion of an oscillating weight suspended by a spring was measured by a motion detector. The data collected and the approximate maximum (positive and negative) displacements from equilibrium are shown in the figure. The displacement is measured in feet and the time is measured in seconds. (a) Is a function of Explain. (b) Approximate the amplitude and period of the oscillations. (c) Find a model for the data. (d) Use a graphing utility to graph the model in part (c). Compare the result with the data in the figure. xy4

Consider the circle as shown in the figure. (a) Find the center and radius of the circle. (b) Find an equation of the tangent line to the circle at the point (c) Find an equation of the tangent line to the circle at the point (d) Where do the two tangent lines intersect? y4x

There are two tangent lines from the point to the circle (see figure). Find equations of these two lines by using the fact that each tangent line intersects the circle in exactly one point. y4x

The Heaviside function is widely used in engineering applications. Sketch the graph of the Heaviside function and the graphs of the following functions by hand. 4x3

Consider the graph of the function shown below. Use this graph to sketch the graphs of the following functions. To print an enlarged copy of the graph, select the MathGraph button. 4x3

A rancher plans to fence a rectangular pasture adjacent to a river. The rancher has 100 meters of fence, and no fencing is needed along the river (see figure). (a) Write the area of the pasture as a function of the length of the side parallel to the river. What is the domain of (b) Graph the area function and estimate the dimensions that yield the maximum amount of area for the pasture. (c) Find the dimensions that yield the maximum amount of area for the pasture by completing the square. x3y

A rancher has 300 feet of fence to enclose two adjacent pastures. (a) Write the total area of the two pastures as a function of (see figure). What is the domain of (b) Graph the area function and estimate the dimensions that yield the maximum amount of area for the pastures. (c) Find the dimensions that yield the maximum amount of area for the pastures by completing the square. 3y2

You are in a boat 2 miles from the nearest point on the coast. You are to go to a point located 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and walk at 4 miles per hour. Write the total time of the trip as a function of 3y2x

You drive to the beach at a rate of 120 kilometers per hour. On the return trip, you drive at a rate of 60 kilometers per hour. What is your average speed for the entire trip? Explain your reasoning. y2x2

One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point on the graph of (a) Find the slope of the line joining and Is the slope of the tangent line at greater than or less than this number? (b) Find the slope of the line joining and Is the slope of the tangent line at greater than or less than this number? (c) Find the slope of the line joining and Is the slope of the tangent line at greater than or less than this number? (d) Find the slope of the line joining and in terms of the nonzero number Verify that and 0.1 yield the solutions to parts (a)(c) above. (e) What is the slope of the tangent line at Explain how you arrived at your answer. y2x2

Sketch the graph of the function and label the point on the graph. (a) Find the slope of the line joining and Is the slope of the tangent line at greater than or less than this number? (b) Find the slope of the line joining and . Is the slope of the tangent line at greater than or less than this number? (c) Find the slope of the line joining and Is the slope of the tangent line at greater than or less than this number? (d) Find the slope of the line joining and in terms of the nonzero number (e) What is the slope of the tangent line at the point Explain how you arrived at your answer. 2x2y

A large room contains two speakers that are 3 meters apart. The sound intensity of one speaker is twice that of the other, as shown in the figure. (To print an enlarged copy of the graph, select the MathGraph button.) Suppose the listener is free to move about the room to find those positions that receive equal amounts of sound from both speakers. Such a location satisfies two conditions: (1) the sound intensity at the listeners position is directly proportional to the sound level of a source, and (2) the sound intensity is inversely proportional to the square of the distance from the source. (a) Find the points on the -axis that receive equal amounts of sound from both speakers. (b) Find and graph the equation of all locations where one could stand and receive equal amounts of sound from both speakers. x2y

Suppose the speakers in Exercise 11 are 4 meters apart and the sound intensity of one speaker is times that of the other, as shown in the figure. To print an enlarged copy of the graph, select the MathGraph button. (a) Find the equation of all locations where one could stand and receive equal amounts of sound from both speakers. (b) Graph the equation for the case (c) Describe the set of locations of equal sound as becomes very large. 2y6

Let and be the distances from the point to the points and respectively, as shown in the figure. Show that the equation of the graph of all points satisfying is This curve is called a lemniscate. Graph the lemniscate and identify three points on the graph. y65

Let (a) What are the domain and range of (b) Find the composition What is the domain of this function? (c) Find What is the domain of this function? (d) Graph Is the graph a line? Why or why not? y65x