Minimum Length The wall of a building is to be braced by a beam that must pass over a

Give the definition of a critical number, and graph a function showing the different types of critical numbers. 2y

Consider the odd function that is continuous and differentiable and has the functional values shown in the table. (a) Determine (b) Determine (c) Plot the points and make a possible sketch of the graph of on the interval What is the smallest number of critical points in the interval? Explain. (d) Does there exist at least one real number in the interval where Explain. (e) Is it possible that does not exist? Explain. (f) Is it necessary that exists at Explain 2y

In Exercises 3 and 4, find the absolute extrema of the function on the closed interval. Use a graphing utility to graph the function over the given interval to confirm your results. yx

In Exercises 3 and 4, find the absolute extrema of the function on the closed interval. Use a graphing utility to graph the function over the given interval to confirm your results. yx

In Exercises 5 and 6, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that x3

In Exercises 5 and 6, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that x3y

Consider the function (a) Graph the function and verify that (b) Note that is not equal to zero for any in Explain why this does not contradict Rolles Theorem. x3y

Can the Mean Value Theorem be applied to the function on the interval Explain. 3y

In Exercises 912, find the point(s) guaranteed by the Mean Value Theorem for the closed interval 3yx

In Exercises 912, find the point(s) guaranteed by the Mean Value Theorem for the closed interval yx3

In Exercises 912, find the point(s) guaranteed by the Mean Value Theorem for the closed interval x3

In Exercises 912, find the point(s) guaranteed by the Mean Value Theorem for the closed interval x3x

For the function determine the value of guaranteed by the Mean Value Theorem on the interval x3x

Demonstrate the result of Exercise 13 for on the interval 3xy

In Exercises 1518, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. xy

In Exercises 1518, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. xyx

In Exercises 1518, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. yx2

In Exercises 1518, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. yx2x

In Exercises 19 and 20, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to verify your results. x2x2

In Exercises 19 and 20, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to verify your results. x2x2

Harmonic Motion The height of an object attached to a spring is given by the harmonic equation where is measured in inches and is measured in seconds. (a) Calculate the height and velocity of the object when second. (b) Show that the maximum displacement of the object is inch. (c) Find the period of Also, find the frequency (number of oscillations per second) if 2x21

Writing The general equation giving the height of an oscillating object attached to a spring is where is the spring constant and is the mass of the object. (a) Show that the maximum displacement of the object is (b) Show that the object oscillates with a frequency of x21y

In Exercises 23 and 24, determine the points of inflection and discuss the concavity of the graph of the function. 21y

In Exercises 23 and 24, determine the points of inflection and discuss the concavity of the graph of the function. 1yx

In Exercises 25 and 26, use the Second Derivative Test to find all relative extrema. 1yxx

In Exercises 25 and 26, use the Second Derivative Test to find all relative extrema. yxx2

Think About It In Exercises 27 and 28, sketch the graph of a function having the given characteristics. xx2

Think About It In Exercises 27 and 28, sketch the graph of a function having the given characteristics. xx21

Writing A newspaper headline states that The rate of growth of the national deficit is decreasing. What does this mean? What does it imply about the graph of the deficit as a function of time? x21xy

Inventory Cost The cost of inventory depends on the ordering and storage costs according to the inventory model Determine the order size that will minimize the cost, assuming that sales occur at a constant rate, is the number of units sold per year, is the cost of storing one unit for 1 year, is the cost of placing an order, and is the number of units per order. 21xy

Modeling Data Outlays for national defense (in billions of dollars) for selected years from 1970 through 1999 are shown in the table, where is time in years, with corresponding to 1970. (Source: U.S. Office of Management and Budget) (a) Use the regression capabilities of a graphing utility to fit a model of the form to the data. (b) Use a graphing utility to plot the data and graph the model. (c) For the years shown in the table, when does the model indicate that the outlay for national defense is at a maximum? When is it at a minimum? (d) For the years shown in the table, when does the model indicate that the outlay for national defense is increasing at the greatest rate? 1xy

Modeling Data The manager of a store recorded the annual sales (in thousands of dollars) of a product over a period of 7 years, as shown in the table, where is the time in years, with corresponding to 1997. (a) Use the regression capabilities of a graphing utility to find a model of the form for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use calculus to find the time when sales were increasing at the greatest rate. (d) Do you think the model would be accurate for predicting future sales? Explain. 1xy4

In Exercises 33 40, find the limit. xy4

In Exercises 33 40, find the limit. 4x

In Exercises 33 40, find the limit. 4x2

In Exercises 33 40, find the limit. 4x2

In Exercises 33 40, find the limit. x20

In Exercises 33 40, find the limit. x20y

In Exercises 33 40, find the limit. 20y

In Exercises 33 40, find the limit. 0y4

In Exercises 4144, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. 0y4

In Exercises 4144, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. y4

In Exercises 4144, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. 4x

In Exercises 4144, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. 4x

In Exercises 4548, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. x3

In Exercises 4548, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. x3xy

In Exercises 4548, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. x3xy2

In Exercises 4548, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. 3xy2

In Exercises 4966, analyze and sketch the graph of the function. 3xy210

In Exercises 4966, analyze and sketch the graph of the function. xy210xy

In Exercises 4966, analyze and sketch the graph of the function. 210xy

In Exercises 4966, analyze and sketch the graph of the function. 10xy4

In Exercises 4966, analyze and sketch the graph of the function. 10xy4y

In Exercises 4966, analyze and sketch the graph of the function. xy4y

In Exercises 4966, analyze and sketch the graph of the function. 4yx

In Exercises 4966, analyze and sketch the graph of the function. 4yx3

In Exercises 4966, analyze and sketch the graph of the function. yx3

In Exercises 4966, analyze and sketch the graph of the function. x3x

In Exercises 4966, analyze and sketch the graph of the function. x3xy

In Exercises 4966, analyze and sketch the graph of the function. 3xy2

In Exercises 4966, analyze and sketch the graph of the function. xy2

In Exercises 4966, analyze and sketch the graph of the function. xy2x

In Exercises 4966, analyze and sketch the graph of the function. y2x3

In Exercises 4966, analyze and sketch the graph of the function. 2x3

In Exercises 4966, analyze and sketch the graph of the function. x34

In Exercises 4966, analyze and sketch the graph of the function. x34x

Find the maximum and minimum points on the graph of (a) without using calculus. (b) using calculus. 34xy

Consider the function for positive integer values of (a) For what values of does the function have a relative minimum at the origin? (b) For what values of does the function have a point of inflection at the origin? 4xy

Distance At noon, ship is 100 kilometers due east of ship Ship is sailing west at 12 kilometers per hour, and ship is sailing south at 10 kilometers per hour. At what time will the ships be nearest to each other, and what will this distance be? 4xyx

Maximum Area Find the dimensions of the rectangle of maximum area, with sides parallel to the coordinate axes, that can be inscribed in the ellipse given by xyx2

Minimum Length A right triangle in the first quadrant has the coordinate axes as sides, and the hypotenuse passes through the point Find the vertices of the triangle such that the length of the hypotenuse is minimum. yx2

Minimum Length The wall of a building is to be braced by a beam that must pass over a parallel fence 5 feet high and 4 feet from the building. Find the length of the shortest beam that can be used. x2x

Maximum Area Three sides of a trapezoid have the same length Of all such possible trapezoids, show that the one of maximum area has a fourth side of length x2xy

Maximum Area Show that the greatest area of any rectangle inscribed in a triangle is one-half that of the triangle. 2xy

Distance Find the length of the longest pipe that can be carried level around a right-angle corner at the intersection of two corridors of widths 4 feet and 6 feet. (Do not use trigonometry.) xyx

Distance Rework Exercise 75, given corridors of widths meters and meters. xyx2

Distance A hallway of width 6 feet meets a hallway of width 9 feet at right angles. Find the length of the longest pipe that can be carried level around this corner. [Hint: If is the length of the pipe, show that where is the angle between the pipe and the wall of the narrower hallway.] yx2

Length Rework Exercise 77, given that one hallway is of width meters and the other is of width meters. Show that the result is the same as in Exercise 76. x22

Minimum Cost In Exercises 79 and 80, find the speed in miles per hour, that will minimize costs on a 110-mile delivery trip. The cost per hour for fuel is dollars, and the driver is paid dollars per hour. (Assume there are no costs other than wages and fuel.) x22y

Minimum Cost In Exercises 79 and 80, find the speed in miles per hour, that will minimize costs on a 110-mile delivery trip. The cost per hour for fuel is dollars, and the driver is paid dollars per hour. (Assume there are no costs other than wages and fuel.) 22y

In Exercises 81 and 82, use Newtons Method to approximate any real zeros of the function accurate to three decimal places. Use the zero or root feature of a graphing utility to verify your results. 2y2

In Exercises 81 and 82, use Newtons Method to approximate any real zeros of the function accurate to three decimal places. Use the zero or root feature of a graphing utility to verify your results. 2y2x

In Exercises 83 and 84, use Newtons Method to approximate, to three decimal places, the -value(s) of the point(s) of intersection of the equations. Use a graphing utility to verify your results. y2x

In Exercises 83 and 84, use Newtons Method to approximate, to three decimal places, the -value(s) of the point(s) of intersection of the equations. Use a graphing utility to verify your results. 2x

In Exercises 85 and 86, find the differential 2xx

In Exercises 85 and 86, find the differential xx2

Surface Area and Volume The diameter of a sphere is measured to be 18 centimeters, with a maximum possible error of 0.05 centimeter. Use differentials to approximate the possible propagated error and percent error in calculating the surface area and the volume of the sphere. x2

Demand Function A company finds that the demand for its commodity is If changes from 7 to 8, find and compare the values of and dp. x21

Graph the fourth-degree polynomial for various values of the constant (a) Determine the values of for which has exactly one relative minimum. (b) Determine the values of for which has exactly one relative maximum. (c) Determine the values of for which has exactly two relative minima. (d) Show that the graph of cannot have exactly two relative extrema. x21x

(a) Graph the fourth-degree polynomial for 0, 1, 2, and 3. For what values of the constant does have a relative minimum or relative maximum? (b) Show that has a relative maximum for all values of the constant (c) Determine analytically the values of for which has a relative minimum. (d) Let be a relative extremum of Show that lies on the graph of Verify this result graphically by graphing together with the seven curves from part (a). 21x2

Let Determine all values of the constant such that has a relative minimum, but no relative maximum. 1x2y

(a) Let be a quadratic polynomial. How many points of inflection does the graph of have? (b) Let be a cubic polynomial. How many points of inflection does the graph of have? (c) Suppose the function satisfies the equation where and are positive constants. Show that the graph of has a point of inflection at the point where (This equation is called the logistic differential equation.) 1x2y

Prove Darbouxs Theorem: Let be differentiable on the closed interval such that and If lies between and , then there exists in such that x2yx

Let and be functions that are continuous on and differentiable on . Prove that if and for all in then 2yx2

Prove the following Extended Mean Value Theorem. If and are continuous on the closed interval , and if exists in the open interval then there exists a number in such that yx2

(a) Let . Find and Show that for small values of the difference is very small in the sense that there exists such that where as (b) Generalize this result by showing that if is a differentiable function, then where as x24

The amount of illumination of a surface is proportional to the intensity of the light source, inversely proportional to the square of the distance from the light source, and proportional to where is the angle at which the light strikes the surface. A rectangular room measures 10 feet by 24 feet, with a 10-foot ceiling. Determine the height at which the light should be placed to allow the corners of the floor to receive as much light as possible x24y

Consider a room in the shape of a cube, 4 meters on each side. A bug at point wants to walk to point at the opposite corner, as shown in the figure. Use calculus to determine the shortest path. Can you solve the problem without calculus? 24y

The line joining and crosses the two parallel lines, as shown in the figure. The point is units from How far from should the point be chosen so that the sum of the areas of the two shaded triangles is a minimum? So that the sum is a maximum? 4y0

The figures show a rectangle, a circle, and a semicircle inscribed in a triangle bounded by the coordinate axes and the first-quadrant portion of the line with intercepts and Find the dimensions of each inscribed figure such that its area is maximum. State whether calculus was helpful in finding the required dimensions. Explain your reasoning. 4y0y

(a) Prove that (b) Prove that (c) Let be a real number. Prove that if then y0y

Find the point on the graph of (see figure) where the tangent line has the greatest slope, and the point where the tangent line has the least slope. 0yx

(a) Let be a positive number. Use the table feature of a graphing utility to verify that (b) Use the Mean Value Theorem to prove that for all positive real numbers 0yx2

(a) Let be a positive number. Use the table feature of a graphing utility to verify that (b) Use the Mean Value Theorem to prove that for all positive real numbers yx2

The police department must determine the speed limit on a bridge such that the flow rate of cars is maximum per unit time. The greater the speed limit, the farther apart the cars must be in order to keep a safe stopping distance. Experimental data on the stopping distance (in meters) for various speeds (in kilometers per hour) are shown in the table. (a) Convert the speeds in the table to the speeds in meters per second. Use the regression capabilities of a graphing utility to find a model of the form for the data. (b) Consider two consecutive vehicles of average length 5.5 meters, traveling at a safe speed on the bridge. Let be the difference between the times (in seconds) when the front bumpers of the vehicles pass a given point on the bridge. Verify that this difference in times is given by (c) Use a graphing utility to graph the function and estimate the speed that minimizes the time between vehicles. (d) Use calculus to determine the speed that minimizes What is the minimum value of Convert the required speed to kilometers per hour. (e) Find the optimal distance between vehicles for the posted speed limit determined in part (d). x23

A legal-sized sheet of paper (8.5 inches by 14 inches) is folded so that corner touches the opposite 14 inch edge at Note: (a) Show that (b) What is the domain of (c) Determine the -value that minimizes (d) Determine the minimum length x23x

The polynomial is the quadratic approximation of the function at if and (a) Find the quadratic approximation of at (b) Use a graphing utility to graph and in the same viewing window. 23x