fx x 2x 3 3, 2 2

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

Consider the odd function f that is continuous and differentiable and has the functional values shown in the table. (a) Determine f(4). (b) Determine f(-3). (c) Plot the points and make a possible sketch of the graph of f on the interval [-6, 6]. What is the smallest number of critical points in the interval? Explain. (d) Does there exist at least one real number c in the interval (-6,6) where f’(c) = -1? Explain. (e) Is it possible that \(\lim _{x \rightarrow 0} f(x)\) does not exist? Explain. (f) Is it necessary that f’(x) exists at x = 2? Explain. Text Transcription: lim_x rightarrow 0 f(x)

In Exercises 3-6, 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. \(f(x)=x^{2}+5 x, \quad[-4,0]\) Text Transcription: f(x)=x^2+5 x, [-4,0]

In Exercises 3-6, 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. \(h(x)=3 \sqrt{x}-x, \quad[0,9]\) Text Transcription: h(x)=3 sqrt x-x, [0, 9]f(x)=x^2+5 x, [-4,0]

In Exercises 3-6, 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. \(g(x)=2 x+5 \cos x, \quad[0,2 \pi]\) Text Transcription: g(x)=2x+5 cos x, [0, 2 pi]

In Exercises 3-6, 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. \(f(x)=\frac{x}{\sqrt{x^{2}+1}}, \quad[0,2]\) Text Transcription: f(x)=x/sqrt x^2+1, [0, 2]

In Exercises 7-10, 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. If Rolle's Theorem cannot be applied, explain why not. \(f(x)=2 x^{2}-7, \quad[0,4]\) Text Transcription: f(x)=2 x^2-7, [0,4]

In Exercises 7-10, 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. If Rolle's Theorem cannot be applied, explain why not. \(f(x)=(x-2)(x+3)^{2}, \quad[-3,2]\) Text Transcription: f(x)=(x-2)(x+3)^2, [-3,2]

In Exercises 7-10, 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. If Rolle's Theorem cannot be applied, explain why not. \(f(x)=\frac{x^{2}}{1-x^{2}}, \quad[-2,2]\) Text Transcription: f(x)=x^2/1-x^2, [-2, 2]

In Exercises 7-10, 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. If Rolle's Theorem cannot be applied, explain why not. \(f(x)=|x-2|-2, \quad[0,4]\) Text Transcription: f(x)=|x-2|-2, [0, 4]

Consider the function f(x) = 3 |x 4|. (a) Graph the function and verify that f(1) = f(7). (b) Note that f'(x) is not equal to zero for any x in [1, 7]. Explain why this does not contradict Rolle's Theorem.

Can the Mean Value Theorem be applied to the function \(f(x)=1 / x^{2}\) on the interval [-2, 1]? Explain. Text Transcription: f(x)=1/x^2

In Exercises 13-18, determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that \(f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\) If the Mean Value Theorem cannot be applied, explain why not. \(f(x)=x^{2 / 3}, \quad[1,8]\) Text Transcription: f’(c)=f(b)-f(a)/b-a f(x)=x^2/3, [1,8]

In Exercises 13-18, determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that \(f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\) If the Mean Value Theorem cannot be applied, explain why not. \(f(x)=\frac{1}{x}, \quad[1,4]\) Text Transcription: f’(c)=f(b)-f(a)/b-a f(x)=1/x, [1, 4]

In Exercises 13-18, determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that \(f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\) If the Mean Value Theorem cannot be applied, explain why not. \(f(x)=|5-x|, \quad[2,6]\) Text Transcription: f’(c)=f(b)-f(a)/b-a f(x)=|5-x|,[2,6]

In Exercises 13-18, determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that \(f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\) If the Mean Value Theorem cannot be applied, explain why not. \(f(x)=2 x-3 \sqrt{x}, \quad[-1,1]\) Text Transcription: f’(c)=f(b)-f(a)/b-a f(x)=2 x-3 sqrt x, [-1,1]

In Exercises 13-18, determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that \(f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\) If the Mean Value Theorem cannot be applied, explain why not. \(f(x)=x-\cos x, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) Text Transcription: f’(c)=f(b)-f(a)/b-a f(x)=x-cos x, [-pi/2, pi/2]

In Exercises 13-18, determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that \(f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\) If the Mean Value Theorem cannot be applied, explain why not. \(f(x)=\sqrt{x}-2 x, \quad[0,4]\) Text Transcription: f’(c)=f(b)-f(a)/b-a f(x)=sqrt x-2 x, [0,4]

For the function \(f(x)=A x^{2}+B x+C\), determine the value of c guaranteed by the Mean Value Theorem on the interval \(\left[x_{1}, x_{2}\right]\). Text Transcription: f(x)=Ax^2+Bx+C [x_1, x_2]

Demonstrate the result of Exercise 19 for \(f(x)=2 x^{2}-3 x+1\) on the interval [0, 4]. Text Transcription: f(x)=2x^2-3x+1

In Exercises 21-26, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. \(f(x)=x^{2}+3 x-12\) Text Transcription: f(x)=x^2+3 x-12

In Exercises 21-26, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. \(h(x)=(x+2)^{1 / 3}+8\) Text Transcription: h(x)=(x+2)^1/ 3+8

In Exercises 21-26, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. \(f(x)=(x-1)^{2}(x-3)\) Text Transcription: f(x)=(x-1)^2(x-3)

In Exercises 21-26, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. \(g(x)=(x+1)^{3}\) Text Transcription: g(x)=(x+1)^3

In Exercises 21-26, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. \(h(x)=\sqrt{x}(x-3), \quad x>0\) Text Transcription: h(x)=sqrt x(x-3), x>0

In Exercises 21-26, find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. \(f(x)=\sin x+\cos x, \quad[0,2 \pi]\) Text Transcription: f(x)=sin x+cos x, [0,2 pi]

In Exercises 27-30, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to confirm your results. \(f(x)=4 x^{3}-5 x\) Text Transcription: f(x)=4x^3-5x

In Exercises 27-30, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to confirm your results. \(g(x)=\frac{x^{3}-8 x}{4}\) Text Transcription: g(x)=x^3-8x/4

In Exercises 27-30, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to confirm your results. \(h(t)=\frac{1}{4} t^{4}-8 t\) Text Transcription: h(t)=1/4t^4-8t

In Exercises 27-30, use the First Derivative Test to find any relative extrema of the function. Use a graphing utility to confirm your results. \(g(x)=\frac{3}{2} \sin \left(\frac{\pi x}{2}-1\right), \quad[0,4]\) 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 \(y=\frac{1}{3} \cos 12 t-\frac{1}{4} \sin 12 t\) where y is measured in inches and t is measured in seconds. (a) Calculate the height and velocity of the object when \(t=\pi / 8\) second. (b) Show that the maximum displacement of the object is \(\frac{5}{12}\) inch. (c) Find the period P of y. Also, find the frequency f (number of oscillations per second) if f = 1/P. Text Transcription: y=1/3 cos 12t-1/4sin 12t t=pi/8

Writing The general equation giving the height of an oscillating object attached to a spring is \(y=A \sin \sqrt{\frac{k}{m}} t+B \cos \sqrt{\frac{k}{m}} t\) where k is the spring constant and m is the mass of the object. (a) Show that the maximum displacement of the object is \(\sqrt{A^{2}+B^{2}}\)? (b) Show that the object oscillates with a frequency of \(f=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}\). Text Transcription: y=A sin sqrt k/m t + B cos sqrt k/m t sqrt A^2+B^2 f=1/2pi sqrt k/m

In Exercises 33-36, determine the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x^{3}-9 x^{2}\) Text Transcription: f(x)=x^3-9 x^2

In Exercises 33-36, determine the points of inflection and discuss the concavity of the graph of the function. \(g(x)=x \sqrt{x+5}\) Text Transcription: g(x)=x sqrt{x+5}

In Exercises 33-36, determine the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x+\cos x, \quad[0,2 \pi]\) Text Transcription: f(x)=x+cos x,[0,2 pi]

In Exercises 33-36, determine the points of inflection and discuss the concavity of the graph of the function. \(f(x)=(x+2)^{2}(x-4)\) Text Transcription: f(x)=(x+2)^2(x-4)

In Exercises 37-40, use the Second Derivative Test to find all relative extrema. \(f(x)=(x+9)^{2}\) Text Transcription: f(x)=(x+9)^2

In Exercises 37-40, use the Second Derivative Test to find all relative extrema. \(h(x)=x-2 \cos x, \quad[0,4 \pi]\) Text Transcription: h(x)=x-2 cos x,[0,4 pi]

In Exercises 37-40, use the Second Derivative Test to find all relative extrema. \(g(x)=2 x^{2}\left(1-x^{2}\right)\) Text Transcription: g(x)=2 x^2(1-x^2)

In Exercises 37-40, use the Second Derivative Test to find all relative extrema. \(h(t)=t-4 \sqrt{t+1}\) Text Transcription: h(t)=t-4 sqrt t+1

Think About It In Exercises 41 and 42, sketch the graph of a function f having the given characteristics. \(\begin{array}{l}f(0)=f(6)=0 \\f^{\prime}(3)=f^{\prime}(5)=0 \\f^{\prime}(x)>0 \text { if } x<3 \\f^{\prime}(x)>0 \text { if } 3<x<5 \\f^{\prime}(x)<0 \text { if } x>5 \\f^{\prime \prime}(x)<0 \text { if } x<3 \text { or } x>4 \\f^{\prime \prime}(x)>0 \text { if } 3<x<4\end{array}\) Text Transcription: f(0)=f(6)=0 f’(3)=f’(5)=0 f’(x)>0 if x<3 f’(x)>0 if 3<x<5 f’(x)<0 if x>5 f’’(x)<0 if x<3 or x>4 f’’(x)>0 if 3<x<4

Think About It In Exercises 41 and 42, sketch the graph of a function f having the given characteristics. \(\begin{array}{l}f(0)=4, f(6)=0 \\f^{\prime}(x)<0 \text { if } x<2 \text { or } x>4 \\f^{\prime}(2) \text { does not exist. } \\f^{\prime}(4)=0 \\f^{\prime}(x)>0 \text { if } 2<x<4 \\f^{\prime \prime}(x)<0 \text { if } x \neq 2\end{array}\) Text Transcription: f(0)=4, f(6)=0 f'(x)<0 if x<2 or x>4 f'(2) does not exist. f'(4)=0 f'(x)>0 if 2<x<4 f’’(x)<0 if x neq 2

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 \(C=\left(\frac{Q}{x}\right) s+\left(\frac{x}{2}\right) r\). Determine the order size that will minimize the cost, assuming that sales occur at a constant rate, Q is the number of units sold per year, r is the cost of storing one unit for l year, s is the cost of placing an order, and x is the number of units per order. Text Transcription: C=(Q/x)s+(x/2)r

Modeling Data Outlays for national defense D (in billions of dollars) for selected years from 1970 through 2005 are shown in the table, where t is time in years, with t = 0 corresponding to 1970. (Source: U.S. Office of Management and Budget) Use the regression capabilities of a graphing utility to fit a model of the form \(D=a t^{4}+b t^{3}+c t^{2}+d t+e\) 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 was at a maximum? When was it at a minimum? (d) For the years shown in the table, when does the model indicate that the outlay for national defense was increasing at the greatest rate? Text Transcription: D=at^4+bt^3+ct^2+dt+e

Modeling Data The manager of a store recorded the annual sales Sin thousands of dollars) of a product over a period of 7 years, as shown in the table, where t is the time in years, with t = 1 corresponding to 2001. (a) Use the regression capabilities of a graphing utility to find a model of the form \(S=a t^{3}+b t^{2}+c t+d\) for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use calculus and the model to find the time t when sales were increasing at the greatest rate. (d) Do you think the model would be accurate for predicting future sales? Explain. Text Transcription: S=at^3+bt^2+ct+d

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow \infty}\left(8+\frac{1}{x}\right)\) Text Transcription: lim_x rightarrow infinity (8+1/x)

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow \infty} \frac{3-x}{2 x+5}\) Text Transcription: lim_x rightarrow infinity 3-x/2x+5

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow \infty} \frac{2 x^{2}}{3 x^{2}+5}\) Text Transcription: lim_x rightarrow infinity 2x^2x^2+5

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow \infty} \frac{2 x}{3 x^{2}+5}\) Text Transcription: lim_x rightarrow infinity 2x/3x^2+5

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow-\infty} \frac{3 x^{2}}{x+5}\) Text Transcription: lim_x rightarrow -infinity 3x^2/x+5

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+x}}{-2 x}\) Text Transcription: lim_x rightarrow -infinity sqrt x^2+x/-2x

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow \infty} \frac{5 \cos x}{x}\) Text Transcription: lim_x rightarrow infinity 5 cos x/x

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{x^{2}+4}}\) Text Transcription: lim_x rightarrow infinity 3x/sqrt x^2+4

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow-\infty} \frac{6 x}{x+\cos x}\) Text Transcription: lim_x rightarrow -infinity 6x/x+cos x

In Exercises 47- 56, find the limit \(\lim _{x \rightarrow-\infty} \frac{x}{2 \sin x}\) Text Transcription: lim_x rightarrow -infinity x/2 sin x

In Exercises 57-60, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. \(f(x)=\frac{3}{x}-2\) Text Transcription: f(x)=3/x-2

In Exercises 57-60, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. \(g(x)=\frac{5 x^{2}}{x^{2}+2}\) Text Transcription: g(x)=5x^2/x^2+2

In Exercises 57-60, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. \(h(x)=\frac{2 x+3}{x-4}\) Text Transcription: h(x)=2x+3/x-4

In Exercises 57-60, find any vertical and horizontal asymptotes of the graph of the function. Use a graphing utility to verify your results. \(f(x)=\frac{3 x}{\sqrt{x^{2}+2}}\) Text Transcription: f(x)=3x/sqrt x^2+2

In Exercises 61-64, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. \(f(x)=x^{3}+\frac{243}{x}\) Text Transcription: f(x)=x^3+243/x

In Exercises 61-64, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. \(f(x)=\left|x^{3}-3 x^{2}+2 x\right|\) Text Transcription: f(x)=|x^3-3x^2+2x|

In Exercises 61-64, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. \(f(x)=\frac{x-1}{1+3 x^{2}}\) Text Transcription: f(x)=x-1/1+3x^2

In Exercises 61-64, use a graphing utility to graph the function. Use the graph to approximate any relative extrema or asymptotes. \(g(x)=\frac{\pi^{2}}{3}-4 \cos x+\cos 2 x\) Text Transcription: g(x)=pi^2/3 - 4 cos x + cos 2x

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=4 x-x^{2}\) Text Transcription: f(x)=4 x-x^2

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=4 x^{3}-x^{4}\) Text Transcription: f(x)=4 x^3-x^4

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=x \sqrt{16-x^{2}}\) Text Transcription: f(x)=x sqrt 16-x^2

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=\left(x^{2}-4\right)^{2}\) Text Transcription: f(x)=(x^2-4)^2

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=(x-1)^{3}(x-3)^{2}\) Text Transcription: f(x)=(x-1)^3(x-3)^2

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=(x-3)(x+2)^{3}\) Text Transcription: f(x)=(x-3)(x+2)^3

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=x^{1 / 3}(x+3)^{2 / 3}\) Text Transcription: f(x)=x^1/3(x+3)^2/3

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=(x-2)^{1 / 3}(x+1)^{2 / 3}\) Text Transcription: f(x)=(x-2)^1/3(x+1)^2/3

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=\frac{5-3 x}{x-2}\) Text Transcription: f(x)=5-3x/x-2

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=\frac{2 x}{1+x^{2}}\) Text Transcription: f(x)=2x/1+x^2

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=\frac{4}{1+x^{2}}\) Text Transcription: f(x)=4/1+x^2

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=\frac{x^{2}}{1+x^{4}}\) Text Transcription: f(x)=x^2/1+x^4

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=x^{3}+x+\frac{4}{x}\) Text Transcription: f(x)=x^3+x+4/x

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=x^{2}+\frac{1}{x}\) Text Transcription: f(x)=x^2+1/x

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=\left|x^{2}-9\right|\) Text Transcription: f(x)=|x^2-9|

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=|x-1|+|x-3|\) Text Transcription: f(x)=|x-1|+|x-3|

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=x+\cos x, \quad 0 \leq x \leq 2 \pi\) Text Transcription: f(x)=x+cos x, 0 leq x leq 2 pi

In Exercises 65-82, analyze and sketch the graph of the function. \(f(x)=\frac{1}{\pi}(2 \sin \pi x-\sin 2 \pi x), \quad-1 \leq x \leq 1\) Text Transcription: f(x)=1/pi (2 sin pi x - sin 2 pi x), -1 leq x leq 1

Find the maximum and minimum points on the graph of \(x^{2}+4 y^{2}-2 x-16 y+13=0\) (a) without using calculus. (b) using calculus. Text Transcription: x^2+4 y^2-2 x-16 y+13=0

Consider the function \(f(x)=x^{n}\) for positive integer values of n. (a) For what values of n does the function have a relative minimum at the origin? (b) For what values of n does the function have a point of inflection at the origin? Text Transcription: f(x)=x^n

Distance At noon, ship A is 100 kilometers due east of ship B. Ship A is sailing west at 12 kilometers per hour, and ship B 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 \(\frac{x^{2}}{144}+\frac{y^{2}}{16}=1\) Text Transcription: x^2/144+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 (1, 8). 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 s. 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 the area of the triangle.

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.)

Distance Rework Exercise 91, given corridors of widths a meters and b meters.

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 L is the length of the pipe, show that \(L=6 \csc \theta+9 \csc \left(\frac{\pi}{2}-\theta\right)\) where \(\theta\) is the angle between the pipe and the wall of the narrower hallway.] Text Transcription: L=6 csc theta+9 csc(pi/2-theta) theta

Length Rework Exercise 93, given that one hallway is of width a meters and the other is of width b meters. Show that the result is the same as in Exercise 92.

Minimum Cost In Exercises 95 and 96, 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: \(C=\frac{v^{2}}{600}\) Drive: W = $5 Text Transcription: C = v^2/600

Minimum Cost In Exercises 95 and 96, 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: \(C=\frac{v^{2}}{500}\) Drive: W = $7.50 Text Transcription: C = v^2/500

In Exercises 97 and 98, use Newton's 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. \(f(x)=x^{3}-3 x-1\) Text Transcription: f(x)=x^3-3 x-1

In Exercises 97 and 98, use Newton's 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. \(f(x)=x^{3}+2 x+1\) Text Transcription: f(x)=x^3+2 x+1

In Exercises 99 and 100, use Newton's Method to approximate, to three decimal places, the x-value(s) of the point(s) of intersection of the equations. Use a graphing utility to verify your results. \(y=x^{4}\) y = x + 3 Text Transcription: y=x^4

In Exercises 99 and 100, use Newton's Method to approximate, to three decimal places, the x-value(s) of the point(s) of intersection of the equations. Use a graphing utility to verify your results. \(y=\sin \pi x\) y = 1 - x Text Transcription: y=sin pi x

In Exercises 101 and 102, find the differential dy. \(y=x(1-\cos x)\) Text Transcription: y=x(1-cos x)

In Exercises 101 and 102, find the differential dy. \(y=\sqrt{36-x^{2}}\) Text Transcription: y=sqrt 36-x^2

Surface Area and Volume The diameter of a sphere is measured as 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 \(p=75-\frac{1}{4} x\). If x changes from 7 to 8, find and compare the values of \(\Delta\)p and dp. Text Transcription: p=75-1/4 x Delta