In Exercises 2936, (a) find the critical numbers

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(f(x)=x^{2}+5 x, \quad[-4,0]\) Text Transcription: f(x)=x^2+5x [-4, 0]

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(f(x)=x^{3}+6 x^{2}, \quad[-6,1]\) Text Transcription: f(x)=x^3+6x^2, [-6,1]

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(f(x)=\sqrt{x}-2, \quad[0,4]\) Text Transcription: f(x)=sqrt x-2, [0,4]

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(h(x)=3 \sqrt{x}-x, \quad[0,9]\) Text Transcription: h(x)=3 sqrt x - x, [0,9]

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(f(x)=\frac{4 x}{x^{2}+9}, \quad[-4,4]\) Text Transcription: f(x)=4x/x^2+9 [-4,4]

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(f(x)=\frac{x}{\sqrt{x^{2}+1}}\) Text Transcription: f(x)=x/sqrt x^2+1, [0,2]

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(g(x)=2 x+5 \cos x, \quad[0,2 \pi]\) Text Transcription: g(x)=2x+5 cos x, [0,2 pi]

Finding Extrema on a Closed Interval In Exercises 1-8, find the absolute extrema of the function on the closed interval. \(f(x)=\sin 2 x, \quad[0,2 \pi]\) Text Transcription: f(x)=sin 2x, [0, 2 pi]

Using Rolle's Theorem In Exercises 9-12, 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)=2x^2-7, [0,4]

Using Rolle's Theorem In Exercises 9-12, 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]

Using Rolle's Theorem In Exercises 9-12, 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]

Using Rolle's Theorem In Exercises 9-12, 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. \(f(x)=\sin 2 x, \quad[-\pi, \pi]\) Text Transcription: f(x)=sin 2x, [-pi,pi]

Using the Mean Value Theorem 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 ofc 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) f(x)=x^2/3, [1,8]

Using the Mean Value Theorem 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 ofc 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. \(f(x)=\frac{1}{x}\) Text Transcription: f’(c)=f(b)-f(a) f(x)=1/x, [1,4]

Using the Mean Value Theorem 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 ofc 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) f(x)=|5-x|, [2,6]

Using the Mean Value Theorem 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 ofc 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) f(x)=2x-3 sqrt x, [-1,1]

Using the Mean Value Theorem 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 ofc 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) f(x)=x-cos x, [-pi/2, pi/2]

Using the Mean Value Theorem 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 ofc 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. \(f(x)=x \log _{2} x, \quad[1,2]\) Text Transcription: f’(c)=f(b)-f(a) f(x)=x log_2 x, [1,2]

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

Using the Mean Value Theorem (a) 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 [\(x_{1}, x_{2}\)]. (b) Demonstrate the result of part (a) for \(f(x)=2 x^{2}-3 x+1\) on the interval [0, 4]. Text Transcription: f(x)=A x^2+B x+C x_1, x_2 f(x)=2 x^2-3 x+1

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify the open intervals on which the function is increasing or decreasing. \(f(x)=x^{2}+3 x-12\) Text Transcription: f(x)=x^2+3x-12

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify 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

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify 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)

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify the open intervals on which the function is increasing or decreasing. \(g(x)=(x+1)^{3}\) Text Transcription: g(x)=(x+1)^3

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify 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

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify 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]

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify the open intervals on which the function is increasing or decreasing. \(f(t)=(2-t) 2^{t}\) Text Transcription: f(t)=(2-t)2^t

Intervals on Which Is Increasing or Decreasing In Exercises 21-28,identify the open intervals on which the function is increasing or decreasing. \(g(x)=2 x \ln x\) Text Transcription: g(x)=2x ln x

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results. \(f(x)=x^{2}-6 x+5\) Text Transcription: f(x)=x^2-6x+5

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results. \(f(x)=4 x^{3}-5 x\) Text Transcription: f(x)=4x^3-5x

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results. \(h(t)=\frac{1}{4} t^{4}-8 t\) Text Transcription: h(t)=1/4 t^4 - 8t

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results. \(g(x)=\frac{x^{3}-8 x}{4}\) Text Transcription: g(x)=x^3-8x/4

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results. \(f(x)=\frac{x+4}{x^{2}}\) Text Transcription: f(x)=x+4/x^2

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results. \(f(x)=\frac{x^{2}-3 x-4}{x-2}\) Text Transcription: f(x)=x^2-3x-4/x-2

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results. \(f(x)=\cos x-\sin x, \quad(0,2 \pi)\) Text Transcription: f(x)=cos x - sin x, (0, 2 pi)

Applying the First Derivative Test In Exercises 29-36, (a) find the critical numbers of f (if any),(b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) 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]

Finding Points of Inflection In Exercises 37-42,find 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^3-9x^2

Finding Points of Inflection In Exercises 37-42,find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=6 x^{4}-x^{2}\) Text Transcription: f(x)=6x^4-x^2

Finding Points of Inflection In Exercises 37-42,find 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

Finding Points of Inflection In Exercises 37-42,find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=3 x-5 x^{3}\) Text Transcription: f(x)=3x-5x^3

Finding Points of Inflection In Exercises 37-42,find 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]

Finding Points of Inflection In Exercises 37-42,find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=\tan \frac{x}{4}, \quad(0,2 \pi)\) Text Transcription: f(x)=tan x/4, (0, 2 pi)

Using the Second Derivative Test In Exercises 43-48, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=(x+9)^{2}\) Text Transcription: f(x)=(x+9)^2

Using the Second Derivative Test In Exercises 43-48, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=2 x^{3}+11 x^{2}-8 x-12\) Text Transcription: f(x)=2 x^3+11 x^2-8 x-12

Using the Second Derivative Test In Exercises 43-48, find all relative extrema. Use the Second Derivative Test where applicable. \(g(x)=2 x^{2}\left(1-x^{2}\right)\) Text Transcription: g(x)=2x^2 (1-x^2)

Using the Second Derivative Test In Exercises 43-48, find all relative extrema. Use the Second Derivative Test where applicable. \(h(t)=t-4 \sqrt{t+1}\) Text Transcription: h(t)=5-4 sqrt t+1

Using the Second Derivative Test In Exercises 43-48, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=2 x+\frac{18}{x}\) Text Transcription: f(x)=2x+18/x

Using the Second Derivative Test In Exercises 43-48, find all relative extrema. Use the Second Derivative Test where applicable. \(h(x)=x-2 \cos x, \quad[0,4 \pi]\) Text Transcription: h(x)=x-2 cos x, [0, 4 pi]

Think About It In Exercises 49 and 50, sketch the graph of a function f having the given characteristics. f(0) = f(6) = 0 f’(3) = f’(5) = 0 f’(x) > 0 for x < 3 f’(x) > 0 for 3 < x < 5 f’(x) < 0 for x > 5 f’’(x) < 0 for x < 3 or x > 4 f’’(x) > 0 for 3 < x < 4

Think About It In Exercises 49 and 50, sketch the graph of a function f having the given characteristics. f(0) = 4,f(6) = 0 f’(x) < 0 for x < 2 or x > 4 f’(2) does not exist. f’(4) = 0 f’(x) > 0 for 2 < x < 4 f’’(x) < 0 for 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 C 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 one 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)

Modeling Data Outlays for national defense D (in billions of dollars) for selected years from 1970 through 2010 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) (a) Use the regression capabilities of a graphing utility to find a model of the form \(D=a t^{4}+b t^{3}+c t^{2}+d t+e\) for 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=a t^4+b t^3+c t^2+d t+e

Climb Rate The time t (in minutes) for a small plane to climb to an altitude of h feet is \(t=50 \log _{10} \frac{18,000}{18,000-h}\) where 18,000 feet is the plane's 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. Text Transcription: t=50 log_10 18,00/18,000-h

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow \infty}\left(8+\frac{1}{x}\right)\) Text Transcription: lim_x rightarrow infinity (8+1/x)

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow-\infty} \frac{1-4 x}{x+1}\) Text Transcription: lim_x rightarrow -infinity 1-4x/x+1

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow \infty} \frac{2 x^{2}}{3 x^{2}+5}\) Text Transcription: lim_x rightarrow infinity 2x^2/3x^2+5

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow \infty} \frac{4 x^{3}}{x^{4}+3}\) Text Transcription: lim_x rightarrow infinity 4x^3/x^4+3

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow-\infty} \frac{3 x^{2}}{x+5}\) Text Transcription: lim_x rightarrow -infinity 3x^2/x+5

Finding a Limit In Exercises 55–64, 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

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow \infty} \frac{5 \cos x}{x}\) Text Transcription: lim_x rightarrow infinity 5 cos x/x

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow \infty} \frac{x^{3}}{\sqrt{x^{2}+2}}\) Text Transcription: lim_x rightarrow infinity x^3/sqrt x^2+2

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow-\infty} \frac{6 x}{x+\cos x}\) Text Transcription: lim_x rightarrow -infinity 6x/x+cos x

Finding a Limit In Exercises 55–64, find the limit. \(\lim _{x \rightarrow-\infty} \frac{x}{2 \sin x}\) Text Transcription: lim_x rightarrow -infinity x/2 sin x

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(f(x)=\frac{3}{x}-2\) Text Transcription: f(x)=3/x - 2

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(g(x)=\frac{5 x^{2}}{x^{2}+2}\) Text Transcription: g(x)=5x^2/x^2+2

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(h(x)=\frac{2 x+3}{x-4}\) Text Transcription: h(x)=2x+3/x-4

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(f(x)=\frac{3 x}{\sqrt{x^{2}+2}}\) Text Transcription: f(x)=3x/sqrt x^2+2

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(f(x)=\frac{5}{3+2 e^{-x}}\) Text Transcription: f(x)=5/3+2e^-x

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(g(x)=30 x e^{-2 x}\) Text Transcription: g(x)=30xe^-2x

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(g(x)=3 \ln \left(1+e^{-x / 4}\right)\) Text Transcription: g(x)=3 ln(1+e^-x/4)

Horizontal Asymptotes In Exercises 65-72,use a graphing utility to graph the function and identify any horizontal asymptotes. \(h(x)=10 \ln \left(\frac{x}{x+1}\right)\) Text Transcription: h(x)=10 ln(x/x+1)

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=4 x-x^{2}\) Text Transcription: f(x)=4x-x^2

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=4 x^{3}-x^{4}\) Text Transcription: f(x)=4x^3-x^4

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=x \sqrt{16-x^{2}}\) Text Transcription: f(x)=x sqrt 16-x^2

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=\left(x^{2}-4\right)^{2}\) Text Transcription: f(x)=(x^2-4)^2

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=x^{1 / 3}(x+3)^{2 / 3}\) Text Transcription: f(x)=x^1/3 (x+3)^2/3

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=(x-3)(x+2)^{3}\) Text Transcription: f(x)=(x-3)(x+2)^3

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=\frac{5-3 x}{x-2}\) Text Transcription: f(x)=5-3x/x-2

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=\frac{2 x}{1+x^{2}}\) Text Transcription: f(x)=2x/1+x^2

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=x^{3}+x+\frac{4}{x}\) Text Transcription: f(x)=x^3+x+4/x

Analyzing the Graph of a Function In Exercises 73-82, analyze and sketch a graph of the function. Label any intercepts, relative extrema,points of inflection,and asymptotes. Use a graphing utility to verify your results. \(f(x)=x^{2}+\frac{1}{x}\) Text Transcription: f(x)=x^2+1/x

Maximum Area A rancher has 400 feet of fencing with which to enclose two adjacent rectangular corrals (see figure). What dimensions should be used so that the enclosed area will be a maximum?

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.

Modeling Data A meteorologist measures the atmospheric pressure P (in kilograms per square meter) at altitude h in kilometers). The data are shown below. (a) Use a graphing utility to plot the points (h, In P). Use the regression capabilities of the graphing utility to find a linear model for the revised data points. (b) The line in part (a) has the form In P = ah + b. Write the equation in exponential form. (c) Use a graphing utility to plot the original data and graph the exponential model in part (b). (d) Find the rate of change of the pressure when h = 5 and h = 18.

Using a Function 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

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.

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 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

Maximum Volume Find the volume of the largest right circular cone that can be inscribed in a sphere of radius r.

Maximum Volume Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius r.

Comparing \(\Delta y\) and dy In Exercises 93 and 94, use the information to evaluate and compare \(\Delta y\)and dy. Function x-Value Differential of x \(y=0.5 x^{2}\) x = 3 \(\Delta x=d x=0.01\) Text Transcription: Delta y y=0.5 x^2 Delta x=dx=0.01

Comparing \(\Delta y\) and dy In Exercises 93 and 94, use the information to evaluate and compare \(\Delta y\)and dy. Function x-Value Differential of x \(y=x^{3}-6 x\) x = 2 \(\Delta x=d x=0.1\) Text Transcription: Delta y y=x^3-6x Delta x=dx=0.1

Finding a Differential In Exercises 95 and 96,find the differential of the given function. \(y=x(1-\cos x)\) Text Transcription: y=x(1-cos x)

Finding a Differential In Exercises 95 and 96,find the differential of the given function. \(y=\sqrt{36-x^{2}}\) Text Transcription: y=sqrt 36-x^2

Volume and Surface Area The radius of a sphere is measured as 9 centimeters,with a possible error of 0.025 centimeter. (a) Use differentials to approximate the possible propagated error in computing the volume of the sphere. (b) Use differentials to approximate the possible propagated error in computing the surface area of the sphere. (c)Approximate the percent errors in parts (a) and (b).

Demand Function A company finds that the demand for its commodity is \(p=75-\frac{1}{4} x\) where p is the price in dollars and x is the number of units. Find and compare the values of \(\Delta p\) and dp as x changes from 7 to 8 Text Transcription: p=75-1/4 x Delta p

Profit The profit P for a company is \(P=100 x e^{-x / 400}\), where x is sales. Approximate the change and percent change in profit as sales increase from x = 115 to x = 120 units. Text Transcription: P=100xe^-x/400