Solved: In Exercises 5 and 6, determine whether Rolles Theorem can be applied to f on

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

Consider the odd function that is continuous, 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

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 indicated interval to confirm your results. \(g(x)=2 x+5 \cos x,[0,2 \pi]\) Equation Transcription: Text Transcription: g(x) = 2x + 5 cos x, [0, 2 pi]

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 indicated interval to confirm your results. \(f(x)=\frac{x}{\sqrt{x^{2}+1}},[0,2]\) Equation Transcription: Text Transcription: f(x) = x / sqrt x^2 + 1, [0, 2]

In Exercises 5 and 6, determine whether Rolle’s Theorem can be applied to f on the closed interval [a, b]. If Rolle’s Theorem can be applied, find all values of c in the open interval (a, b) such that f ‘(c) = 0. \(f(x)=(x-2)(x+3)^{2},[-3,2]\) Equation Transcription: Text Transcription: f(x) = (x - 2)(x + 3)^2, [-3, 2]

In Exercises 5 and 6, determine whether Rolle’s Theorem can be applied to f on the closed interval [a, b]. If Rolle’s Theorem can be applied, find all values of c in the open interval (a, b) such that f ‘(c) = 0. f(x) = |x - 2| - 2, [0, 4]

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

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

In Exercises 9–12, find the point(s) guaranteed by the Mean Value Theorem for the closed interval [a, b]. \(f(x)=x^{2 / 3},[1,8]\) Equation Transcription: Text Transcription: f(x) = x^2/3, [1, 8]

In Exercises 9–12, find the point(s) guaranteed by the Mean Value Theorem for the closed interval [a, b]. \(f(x)=\frac{1}{x},[1,4]\) Equation Transcription: Text Transcription: f(x) = 1/x, [1, 4]

In Exercises 9–12, find the point(s) guaranteed by the Mean Value Theorem for the closed interval [a, b]. \(f(x)=x-\cos x,\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) Equation Transcription: Text Transcription: f(x) = x - cos x, [- pi/2, pi/2]

In Exercises 9–12, find the point(s) guaranteed by the Mean Value Theorem for the closed interval [a, b]. \(f(x)=x \log _{2} x,[1,2]\) Equation Transcription: Text Transcription: f(x) = x log_2 x, [1,2]

For the function determine the value of guaranteed by the Mean Value Theorem in the intervalx1, x2.

Demonstrate the result of Exercise 13 for on the interval 0, 4.

In Exercises 1520, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasingf
x

x 12
x 3 0,

In Exercises 1520, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasingg
x

x 13 f

In Exercises 1520, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasingh
x
x
x 3, x > 0 g
x

In Exercises 1520, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasingf
x
sin x cos x, 0, 2 h

In Exercises 1520, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasingf
t

2 t2 g
x

In Exercises 1520, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasingg
x
2x ln x t

In Exercises 21 and 22, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to verify your results. \(h(t)=\frac{1}{4} t^{4}-8 t\) Equation Transcription: Text Transcription: h(t) = 1/4 t^4 - 8t

In Exercises 21 and 22, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to verify your results. \(g(x)=\frac{3}{2} \sin \left(\frac{\pi x}{2}-1\right),[0,4]\) Equation Transcription: Text Transcription: g(x) = 3/2 sin (pi x/ 2 - 1), [0, 4]

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 f
1P.

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 f
1 2k m.

In Exercises 25 and 26, determine the points of inflection of the function.f
x
x cos x, 0, 2 f

In Exercises 25 and 26, determine the points of inflection of the function.f
x

x 22
x 4 f
x

In Exercises 27 and 28, use the Second Derivative Test to find all relative extrema.g
x
2x2
1 x2

In Exercises 27 and 28, use the Second Derivative Test to find all relative extrema.h
t
t 4t 1 g

Think About It In Exercises 29 and 30, sketch the graph of a function f having the indicated characteristics.f
x > 0, 3 < x < 4 f
x < 0 x < 3 x > 4 f
x < 0 x > 5 f
x > 0 3 < x < 5 f
x > 0 x < 3 f
3
f
5
0 f
0
f
6

Think About It In Exercises 29 and 30, sketch the graph of a function f having the indicated characteristics.2 f
x > 0 2 < x < 4 f
4
0 f
2 f
x < 0 x < 2 x > 4 f
0

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?

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.

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?

Climb Rate The time (in minutes) for a small plane to climb to an altitude of feet is where 18,000 feet is the planes absolute ceiling. (a) Determine the domain of the function appropriate for the context of the problem. (b) Use a graphing utility to graph the time function and identify any asymptotes. (c) Find the time when the altitude is increasing at the greatest rate.

In Exercises 35 42, find the limit.lim x 2x2 3x2 5

In Exercises 35 42, find the limit.lim x 2x 3x2 5 l

In Exercises 35 42, find the limit.lim x 3x2 x 5 l

In Exercises 35 42, find the limit.im x x2 x 2x lim

In Exercises 35 42, find the limit.lim x 5 cos x x

In Exercises 35 42, find the limit.im x 3x x2 4 l

In Exercises 35 42, find the limit.lim x 6x x cos x l

In Exercises 35 42, find the limit.lim x x 2 sin x l

In Exercises 4350, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results.h
x
2x 3 x 4 li

In Exercises 43–50, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. \(g(x)=\frac{5 x^{2}}{x^{2}+2}\) Equation Transcription: Text Transcription: g(x) = 5x^2 / x^2 + 2

In Exercises 43–50, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. \(g(x)=\frac{3}{x}-2\) Equation Transcription: Text Transcription: g(x) = 3/x - 2

In Exercises 4350, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results.f
x
3x x2 2 f
x

In Exercises 4350, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results.f
x
5 3 2ex f

In Exercises 4350, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results.g
x
30xe2x f

In Exercises 4350, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results.g
x
3 ln
1 ex4 g
x

In Exercises 4350, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results.h
x
10 ln x x 1 g

In Exercises 5154, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes.f
x
x3 243 x h

In Exercises 5154, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes.f
x
x3 3x2 2x f

In Exercises 5154, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes.

In Exercises 5154, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes.g
x
2 3 f
x
4 cos x cos 2x x 1

In Exercises 5580, analyze and sketch the graph of the function.f
x
4x x2

In Exercises 55–80, analyze and sketch the graph of the function. \(f(x)=4 x^{3}-x^{4}\) Equation Transcription: Text Transcription: f(x) = 4x^3 - x^4

In Exercises 55–80, analyze and sketch the graph of the function. \(f(x)=x \sqrt{16-x^{2}}\) Equation Transcription: Text Transcription: f(x) = x sqrt 16 - x^2

In Exercises 5580, analyze and sketch the graph of the function.23 f
x

x 13
x 32 f
x

x

In Exercises 55–80, analyze and sketch the graph of the function. \(f(x)=(x-1)^{3}(x-3)^{2}\) Equation Transcription: Text Transcription: f(x) = (x - 1)^3 (x - 3)^2

In Exercises 55–80, analyze and sketch the graph of the function. \(f(x)=(x-3)(x+2)^{3}\) Equation Transcription: Text Transcription: f(x) = (x - 3) (x + 2)^3

In Exercises 5580, analyze and sketch the graph of the function.f
x
x13
x 323 f

In Exercises 5580, analyze and sketch the graph of the function.f
x

x 213
x 123 f
x

In Exercises 5580, analyze and sketch the graph of the function.f
x
x 1 x 1 28

In Exercises 5580, analyze and sketch the graph of the function.f
x
2x 1 x2 f

In Exercises 55–80, analyze and sketch the graph of the function. \(f(x)=\frac{4}{1+x^{2}}\) Equation Transcription: Text Transcription: f(x) = 4 / 1 + x^2

In Exercises 5580, analyze and sketch the graph of the function.f
x
x2 1 x4 f

In Exercises 5580, analyze and sketch the graph of the function.f
x
x3 x 4 x f

In Exercises 5580, analyze and sketch the graph of the function.f
x
x2 1 x f

In Exercises 55–80, analyze and sketch the graph of the function. \(f(x)=\left|x^{2}-9\right|\) Equation Transcription: Text Transcription: f(x) =|x^2 - 9|

In Exercises 55–80, analyze and sketch the graph of the function. f(x) = |x - 1| + |x - 3|

In Exercises 5580, analyze and sketch the graph of the function.h
x

1 xe x f

In Exercises 55–80, analyze and sketch the graph of the function. \(g(x)=5 x e^{-x^{2}}\) Equation Transcription: Text Transcription: g(x) = 5xe^-x^2

In Exercises 5580, analyze and sketch the graph of the function.g
x

x 3 ln
x 3 g
x

In Exercises 55–80, analyze and sketch the graph of the function. \(h(t)=\frac{l n t}{t^{2}}\) Equation Transcription: Text Transcription: h(t) = ln t / t^2

In Exercises 55–80, analyze and sketch the graph of the function. \(f(x)=\frac{10 \log _{4} x}{x}\) Equation Transcription: Text Transcription: f(x) = 10 log_4 x / x

In Exercises 5580, analyze and sketch the graph of the function.g
x
100x
3x f
x

In Exercises 5580, analyze and sketch the graph of the function.f
x
x cos x, 0 x 2 g

In Exercises 5580, analyze and sketch the graph of the function.f
x
1 x 1 1 
2 sin x sin 2x, f

In Exercises 5580, analyze and sketch the graph of the function.y
4x 6 arctan x

In Exercises 55–80, analyze and sketch the graph of the function. \(y=\frac{1}{2} x^{2}-\arcsin \frac{x}{2}\) Equation Transcription: Text Transcription: y = 1/2 x^2 - arcsin x/2

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

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?

Minimum 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?

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 x2 144 y 2 16
1.

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.

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.

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 2s.

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

Maximum Length 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.)

Maximum Length Rework Exercise 89, given corridors of widths meters and meters.

Maximum Length 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.]

Maximum Length Rework Exercise 91, 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 90.

Minimum Cost In Exercises 93 and 94, find the speed v (in miles per hour) that will minimize costs on a 110-mile delivery trip. The cost per hour for fuel is C dollars, and the driver is paid W dollars per hour. (Assume there are no costs other than wages and fuel.) Fuel Cost = \(C=\frac{v^{2}}{600}\) Driver: W = $5 Equation Transcription: Text Transcription: C = v^2 / 600

Minimum Cost In Exercises 93 and 94, find the speed v (in miles per hour) that will minimize costs on a 110-mile delivery trip. The cost per hour for fuel is C dollars, and the driver is paid W dollars per hour. (Assume there are no costs other than wages and fuel.) Fuel Cost = \(C=\frac{v^{2}}{500}\) Driver: W = $7.50 Equation Transcription: Text Transcription: C = v^2 / 500

In Exercises 95 and 96, find the differential dy.y
x
1 cos x

In Exercises 95 and 96, find the differential dy.y
36 x2 y

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

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 p dp.