Think About It In Exercises 97 and 98, the graph of a position function is shown. It

In Exercises 1 and 2, use the graph to estimate the slope of the tangent line to at the point Verify your answer analytically. To print an enlarged copy of the graph, select the MathGraph button. y2x

In Exercises 1 and 2, use the graph to estimate the slope of the tangent line to at the point Verify your answer analytically. To print an enlarged copy of the graph, select the MathGraph button. 2x

In Exercises 3 24, find the derivative of the function. 23y

In Exercises 3 24, find the derivative of the function. 3y1

In Exercises 3 24, find the derivative of the function. 3y1

In Exercises 3 24, find the derivative of the function. y1x

In Exercises 3 24, find the derivative of the function. 1x2

In Exercises 3 24, find the derivative of the function. 1x2y

In Exercises 3 24, find the derivative of the function. x2y

In Exercises 3 24, find the derivative of the function. x2y2

In Exercises 3 24, find the derivative of the function. 2y23

In Exercises 3 24, find the derivative of the function. y23x

In Exercises 3 24, find the derivative of the function. 23x

In Exercises 3 24, find the derivative of the function. 23x1

In Exercises 3 24, find the derivative of the function. 3x1y

In Exercises 3 24, find the derivative of the function. x1y

In Exercises 3 24, find the derivative of the function. 1y1

In Exercises 3 24, find the derivative of the function. 1y12

In Exercises 3 24, find the derivative of the function. y12x

In Exercises 3 24, find the derivative of the function. 12x

In Exercises 3 24, find the derivative of the function. 12x4

In Exercises 3 24, find the derivative of the function. 2x4y

In Exercises 3 24, find the derivative of the function. x4y

In Exercises 3 24, find the derivative of the function. 4y

In Exercises 2530, complete the table. 4y1

In Exercises 2530, complete the table. y12

In Exercises 2530, complete the table. 12x

In Exercises 2530, complete the table. 12x

In Exercises 2530, complete the table. 12x2

In Exercises 2530, complete the table. 2x2y

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. x2y

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 2y3

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 2y3x

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. y3x

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 3x2

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 3x2

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. x2y

In Exercises 3138, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 2y

In Exercises 3952, find the derivative of the function. 2y

In Exercises 3952, find the derivative of the function. yx

In Exercises 3952, find the derivative of the function. yx

In Exercises 3952, find the derivative of the function. x3

In Exercises 3952, find the derivative of the function. x3y

In Exercises 3952, find the derivative of the function. x3y

In Exercises 3952, find the derivative of the function. 3y

In Exercises 3952, find the derivative of the function. 3yx

In Exercises 3952, find the derivative of the function. yx3

In Exercises 3952, find the derivative of the function. x3

In Exercises 3952, find the derivative of the function. x3x

In Exercises 3952, find the derivative of the function. x3x

In Exercises 3952, find the derivative of the function. 3xy

In Exercises 3952, find the derivative of the function. xy

In Exercises 5356, (a) find an equation of the tangent line to the graph of at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. xyx

In Exercises 5356, (a) find an equation of the tangent line to the graph of at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. yx2

In Exercises 5356, (a) find an equation of the tangent line to the graph of at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. yx2x

In Exercises 5356, (a) find an equation of the tangent line to the graph of at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. x2x2

In Exercises 5762, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. x2x2

In Exercises 5762, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. 2x21

In Exercises 5762, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. x21y

In Exercises 5762, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. 21y

In Exercises 5762, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. 1yx

In Exercises 5762, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. 1yxx

In Exercises 6366, find such that the line is tangent to the graph of the function. yxx2

In Exercises 6366, find such that the line is tangent to the graph of the function. xx2

In Exercises 6366, find such that the line is tangent to the graph of the function. xx21

In Exercises 6366, find such that the line is tangent to the graph of the function. x21xy

Use the graph of to answer each question. To print an enlarged copy of the graph, select the MathGraph button. (a) Between which two consecutive points is the average rate of change of the function greatest? (b) Is the average rate of change of the function between and greater than or less than the instantaneous rate of change at (c) Sketch a tangent line to the graph between and such that the slope of the tangent line is the same as the average rate of change of the function between and 21xy

Sketch the graph of a function such that for all and the rate of change of the function is decreasing. 1xy

In Exercises 69 and 70, the relationship between and is given. Explain the relationship between and 1xy4

In Exercises 69 and 70, the relationship between and is given. Explain the relationship between and xy4

In Exercises 69 and 70, the relationship between and is given. Explain the relationship between and 4x

In Exercises 69 and 70, the relationship between and is given. Explain the relationship between and 4x2

Sketch the graphs of and and sketch the two lines that are tangent to both graphs. Find equations of these lines. 4x2

Show that the graphs of the two equations and have tangent lines that are perpendicular to each other at their point of intersection. x20

Show that the graph of the function does not have a horizontal tangent line. x20y

Show that the graph of the function does not have a tangent line with a slope of 3. 20y

In Exercises 77 and 78, find an equation of the tangent line to the graph of the function through the point not on the graph. To find the point of tangency on the graph of , solve the equation 0y4

In Exercises 77 and 78, find an equation of the tangent line to the graph of the function through the point not on the graph. To find the point of tangency on the graph of , solve the equation 0y4

Linear Approximation Use a graphing utility, with a square window setting, to zoom in on the graph of to approximate Use the derivative to find y4

Linear Approximation Use a graphing utility, with a square window setting, to zoom in on the graph of to approximate Use the derivative to find 4x

Linear Approximation Consider the function with the solution point (a) Use a graphing utility to graph Use the feature to obtain successive magnifications of the graph in the neighborhood of the point After zooming in a few times, the graph should appear nearly linear. Use the feature to determine the coordinates of a point near Find an equation of the secant line through the two points. (b) Find the equation of the line tangent to the graph of passing through the given point. Why are the linear functions and nearly the same? (c) Use a graphing utility to graph and on the same set of coordinate axes. Note that is a good approximation of when is close to 4. What happens to the accuracy of the approximation as you move farther away from the point of tangency? (d) Demonstrate the conclusion in part (c) by completing the table. 4x

Linear Approximation Repeat Exercise 81 for the function where is the line tangent to the graph at the point Explain why the accuracy of the linear approximation decreases more rapidly than in Exercise 81. x3

True or False? In Exercises 8388, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. x3xy

True or False? In Exercises 8388, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. x3xy2

True or False? In Exercises 8388, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 3xy2

True or False? In Exercises 8388, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 3xy210

True or False? In Exercises 8388, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. xy210xy

True or False? In Exercises 8388, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 210xy

In Exercises 8992, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. 10xy4

In Exercises 8992, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. 10xy4y

In Exercises 8992, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. xy4y

In Exercises 8992, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. 4yx

Vertical Motion In Exercises 93 and 94, use the position function for free-falling objects. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average velocity on the interval (c) Find the instantaneous velocities when and (d) Find the time required for the coin to reach ground level. (e) Find the velocity of the coin at impact. 4yx3

Vertical Motion In Exercises 93 and 94, use the position function for free-falling objects. A ball is thrown straight down from the top of a 220-foot building with an initial velocity of feet per second. What is its velocity after 3 seconds? What is its velocity after falling 108 feet? yx3

Vertical Motion In Exercises 95 and 96, use the position function for free-falling objects. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After 10 seconds? x3x

Vertical Motion In Exercises 95 and 96, use the position function for free-falling objects. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen 6.8 seconds after the stone is dropped? x3xy

Think About It In Exercises 97 and 98, the graph of a position function is shown. It represents the distance in miles that a person drives during a 10-minute trip to work. Make a sketch of the corresponding velocity function. 3xy2

Think About It In Exercises 97 and 98, the graph of a position function is shown. It represents the distance in miles that a person drives during a 10-minute trip to work. Make a sketch of the corresponding velocity function. xy2

Think About It In Exercises 99 and 100, the graph of a velocity function is shown. It represents the velocity in miles per hour during a 10-minute drive to work. Make a sketch of the corresponding position function. xy2x

Think About It In Exercises 99 and 100, the graph of a velocity function is shown. It represents the velocity in miles per hour during a 10-minute drive to work. Make a sketch of the corresponding position function. y2x3

Modeling Data The stopping distance of an automobile, on dry, level pavement, traveling at a speed (kilometers per hour) is the distance (meters) the car travels during the reaction time of the driver plus the distance (meters) the car travels after the brakes are applied (see figure). The table shows the results of an experiment. (a) Use the regression capabilities of a graphing utility to find a linear model for reaction time distance. (b) Use the regression capabilities of a graphing utility to find a quadratic model for braking distance. (c) Determine the polynomial giving the total stopping distance (d) Use a graphing utility to graph the functions and in the same viewing window. (e) Find the derivative of and the rates of change of the total stopping distance for and (f) Use the results of this exercise to draw conclusions about the total stopping distance as speed increases. 2x3

Fuel Cost A car is driven 15,000 miles a year and gets miles per gallon. Assume that the average fuel cost is $1.55 per gallon. Find the annual cost of fuel as a function of and use this function to complete the table. Who would benefit more from a one-mile-per-gallon increase in fuel efficiencythe driver of a car that gets 15 miles per gallon or the driver of a car that gets 35 miles per gallon? Explain. x34

Volume The volume of a cube with sides of length is given by Find the rate of change of the volume with respect to when centimeters. x34x

Area The area of a square with sides of length is given by Find the rate of change of the area with respect to when meters. 34xy

Velocity Verify that the average velocity over the time interval is the same as the instantaneous velocity at for the position function 4xy

Inventory Management The annual inventory cost for a manufacturer is where is the order size when the inventory is replenished. Find the change in annual cost when is increased from 350 to 351, and compare this with the instantaneous rate of change when 4xyx

Writing The number of gallons of regular unleaded gasoline sold by a gasoline station at a price of dollars per gallon is given by (a) Describe the meaning of (b) Is usually positive or negative? Explain. xyx2

Newtons Law of Cooling This law states that the rate of change of the temperature of an object is proportional to the difference between the objects temperature and the temperature of the surrounding medium. Write an equation for this law. yx2

Find an equation of the parabola that passes through and is tangent to the line at x2x

Let be an arbitrary point on the graph of Prove that the area of the triangle formed by the tangent line through and the coordinate axes is 2. x2xy

Find the tangent line(s) to the curve through the point 2xy

Find the equation(s) of the tangent line(s) to the parabola through the given point. (a) (b) Are there any restrictions on the constant xyx

In Exercises 113 and 114, find and such that is differentiable everywhere. xyx2

In Exercises 113 and 114, find and such that is differentiable everywhere. yx2

Where are the functions and differentiable? x22

Prove that x22y