Solved: In Exercises 7582, (a) find an equation of the tangent line to the graph of at

In Exercises 1-6, complete the table. y = f(g(x)) u = g(x) y = f(u) \(y=(5x-8)^4\) Text Transcription: y=(5x-8)^4

In Exercises 1-6, complete the table. y = f(g(x)) u = g(x) y = f(u) \(y=\frac{1}{\sqrt{x+1}}\) Text Transcription: y=1/sqrtx+1

In Exercises 1-6, complete the table. y = f(g(x)) u = g(x) y = f(u) \(y=\sqrt{x^3-7}\) Text Transcription: y=sqrtx^3 -7

In Exercises 1-6, complete the table. y = f(g(x)) u = g(x) y = f(u) \(y=3 \tan (\pi x^2)\) Text Transcription: y=3 tan (pi x^2)

In Exercises 1-6, complete the table. y = f(g(x)) u = g(x) y = f(u) \(y=\csc^3 x\) Text Transcription: y=csc^3 x

In Exercises 1-6, complete the table. y = f(g(x)) u = g(x) y = f(u) \(y=\sin \frac{5x}{2}\) Text Transcription: y=sin 5x/2

In Exercises 7–36, find the derivative of the function. \(y=(4x-1)^3\) Text Transcription: y=(4x-1)^3

In Exercises 7–36, find the derivative of the function. \(y=2(6-x^2)^5\) Text Transcription: y=2(6-x^2)^5

In Exercises 7–36, find the derivative of the function. \(g(x)=3(4-9x)^4\) Text Transcription: g(x) = 3(4 - 9x)^4

In Exercises 7–36, find the derivative of the function. \(f(t)=(9t+2)^{2/3}\) Text Transcription: f(t)=(9t+2)^2/3

In Exercises 7–36, find the derivative of the function. \(f(t)=\sqrt{5-t}\) Text Transcription: f(t)=sqrt5-t

In Exercises 7–36, find the derivative of the function. \(g(x)=\sqrt{9-4x}\) Text Transcription: g(x) = sqrt9-4x

In Exercises 7–36, find the derivative of the function. \(y=\sqrt[3]{6x^2+1}\) Text Transcription: y=cube root6x^2 +1

In Exercises 7–36, find the derivative of the function. \(g(x)=\sqrt{x^2-2x+1}\) Text Transcription: g(x)=sqrtx^2 -2x+1

In Exercises 7–36, find the derivative of the function. \(y=2\sqrt[4]{9-x^2}\) Text Transcription: y=2 4th root9-x^2

In Exercises 7–36, find the derivative of the function. \(f(x)=-3\sqrt[4]{2-9x}\) Text Transcription: f(x)=-3 4th root2-9x

In Exercises 7–36, find the derivative of the function. \(y=\frac{1}{x-2}\) Text Transcription: y=1/x-2

In Exercises 7–36, find the derivative of the function. \(s(t)=\frac{1}{t^2+3t-1}\) Text Transcription: s(t)=1/t^2 +3t-1

In Exercises 7–36, find the derivative of the function. \(f(t)=(\frac{1}{t-3})^2\) Text Transcription: f(t)=(1/t-3)^2

In Exercises 7–36, find the derivative of the function. \(y=-\frac{5}{(t-3)^3}\) Text Transcription: y=-5/(t+3)^3

In Exercises 7–36, find the derivative of the function. \(y=\frac{1}{\sqrt{x+2}}\) Text Transcription: y=1/sqrtx+2

In Exercises 7–36, find the derivative of the function. \(g(t)=\sqrt{\frac{1}{t^2-2}}\) Text Transcription: g(t)=sqrt1/t^2 -2

In Exercises 7–36, find the derivative of the function. \(f(x)=x^2(x-2)^4\) Text Transcription: f(x)=x^2(x-2)^4

In Exercises 7–36, find the derivative of the function. \(f(x)=x(3x-9)^3\) Text Transcription: f(x)=x(3x-9)^3

In Exercises 7–36, find the derivative of the function. \(y=x\sqrt{1-x^2}\) Text Transcription: y=xsqrt1-x^2

In Exercises 7–36, find the derivative of the function. \(y=\frac{1}{2}x^2\sqrt{16-x^2}\) Text Transcription: y=1/2 x^2 sqrt16-x^2

In Exercises 7–36, find the derivative of the function. \(y=\frac{x}{\sqrt{x^2+1}}\) Text Transcription: y=x/sqrtx^2 +1

In Exercises 7–36, find the derivative of the function. \(y=\frac{x}{\sqrt{x^4+4}}\) Text Transcription: y=x/sqrtx^4 +4

In Exercises 7–36, find the derivative of the function. \(g(x)=(\frac{x+5}{x^2+2})^2\) Text Transcription: g(x)=(x+5/x^2 +2)^2

In Exercises 7–36, find the derivative of the function. \(h(t)=(\frac{t^2}{t^3+2})^2\) Text Transcription: h(t)=(t^2 /t^3 +2)^2

In Exercises 7–36, find the derivative of the function. \(f(v)=(\frac{1-2v}{1+v})^3\) Text Transcription: f(v)=(1-2v/1+v)^3

In Exercises 7–36, find the derivative of the function. \(g(x)=(\frac{3x^2-2}{2x+3})^3\) Text Transcription: g(x)=(3x^2 -2/2x+3)^3

In Exercises 7–36, find the derivative of the function. \(f(x)=((x^2+3)^5+x)^2\) Text Transcription: f(x)=((x^2 +3)^5 +x)^2

In Exercises 7–36, find the derivative of the function. \(g(x)=(2+(x^2+1)^4)^3\) Text Transcription: g(x)=(2+(x^2 +1)^4 )^3

In Exercises 7–36, find the derivative of the function. \(f(x)=\sqrt{2+\sqrt{2+\sqrt{x}}}\) Text Transcription: f(x)=sqrt2+sqrt2+sqrtx

In Exercises 7–36, find the derivative of the function. \(g(t)=\sqrt{\sqrt{t+1}+1}\) Text Transcription: g(t)=sqrtsqrtt+1 +1

In Exercises 37–42, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \(y=\frac{\sqrt{x}+1}{x^2+1}\) Text Transcription: y=sqrtx +1/x^2 +1

In Exercises 37–42, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \(y=\sqrt{\frac{2x}{x+1}}\) Text Transcription: y=sqrt2x/x+1

In Exercises 37–42, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \(y=\sqrt{\frac{x+1}{x}}\) Text Transcription: y=sqrtx+1/x

In Exercises 37–42, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \(g(x)=\sqrt{x-1}+\sqrt{x+1}\) Text Transcription: g(x)=sqrtx-1 + sqrtx+1

In Exercises 37–42, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \(y=\frac{\cos \pi x +1}{x}\) Text Transcription: y=cos pi x +1/x

In Exercises 37–42, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \(y=x^2 \tan\frac{1}{x}\) Text Transcription: y=x^2 tan 1/x

In Exercises 43 and 44, find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval \([0,\ 2\pi]\). What can you conclude about the slope of the sine function sin ax at the origin? Text Transcription: [0, 2pi]

In Exercises 43 and 44, find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval \([0,\ 2\pi]\). What can you conclude about the slope of the sine function sin ax at the origin? Text Transcription: [0, 2pi]

In Exercises 45–66, find the derivative of the function. y = cos 4x

In Exercises 45–66, find the derivative of the function. \(y=\sin \pi x\) Text Transcription: y = sin pi x

In Exercises 45–66, find the derivative of the function. g(x) = 5 tan 3x

In Exercises 45–66, find the derivative of the function. \(h(x)=\sec x^2\) Text Transcription: h(x)=sec x^2

In Exercises 45–66, find the derivative of the function. \(y=\sin (\pi x)^2\) Text Transcription: y = sin (pi x)^2

In Exercises 45–66, find the derivative of the function. \(y=\cos(1-2x)^2\) Text Transcription: y=cos (1 - 2x)^2

In Exercises 45–66, find the derivative of the function. h(x) = sin 2x cos 2x

In Exercises 45–66, find the derivative of the function. \(g(\theta)=\sec (\frac{1}{2}\theta)\ \tan(\frac{1}{2}\theta)\) Text Transcription: g(theta) = sec (1/2 theta) tan (1/2 theta)

In Exercises 45–66, find the derivative of the function. \(f(x)=\frac{\cot x}{\sin x}\) Text Transcription: f(x)=cot x/ sin x

In Exercises 45–66, find the derivative of the function. \(g(v)=\frac{\cos v}{\csc v}\) Text Transcription: g(v)=cos v/ csc v

In Exercises 45–66, find the derivative of the function. \(y=4 \sec^2 x\) Text Transcription: y=4 sec^2 x

In Exercises 45–66, find the derivative of the function. \(g(t)=5 \cos^2 \pi t\) Text Transcription: g(t)=5 cos^2 pi t

In Exercises 45–66, find the derivative of the function. \(f(\theta)=\tan^2 5 \theta\) Text Transcription: f(theta) = tan^2 5 theta

In Exercises 45–66, find the derivative of the function. \(g(\theta)=\cos^2 8\theta\) Text Transcription: g(theta) = cos^2 8 theta

In Exercises 45–66, find the derivative of the function. \(f(\theta)=\frac{1}{4} \sin^2 2\theta\) Text Transcription: f(theta) = 1/4 sin^2 2 theta

In Exercises 45–66, find the derivative of the function. \(h(t)=2 \cot^2 (\pi t+2)\) Text Transcription: h(t) = 2 cot^2 (pi t + 2)

In Exercises 45–66, find the derivative of the function. \(f(t)=3 \sec^2 (\pi t-1)\) Text Transcription: f(t)=3 sec^2 (pi t - 1)

In Exercises 45–66, find the derivative of the function. \(y=3x-5 \cos (\pi x)^2\) Text Transcription: y=3x-5 cos (pi x)^2

In Exercises 45–66, find the derivative of the function. \(y=\sqrt{x}+\frac{1}{4} \sin(2x)^2\) Text Transcription: y=sqrtx +1/4 sin (2x)^2

In Exercises 45–66, find the derivative of the function. \(y=\sin \sqrt[3]{x}+\sqrt[3]{\sin x}\) Text Transcription: y=sin cube rootx + cube root sin x

In Exercises 45–66, find the derivative of the function. y = sin (tan 2x)

In Exercises 45–66, find the derivative of the function. \(y=\cos \sqrt{\sin(\tan \pi x)}\) Text Transcription: y=cos sqrtsin(tan pi x)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(s(t)=\sqrt{t^2+6t-2},\ \ \ (3,\ 5)\) Text Transcription: s(t)=sqrtt^2 +6t-2, (3, 5)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(y=\sqrt[5]{3x^3+4x},\ \ \ (2,\ 2)\) Text Transcription: y=5th root3x^3 +4x, (2, 2)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(f(x)=\frac{5}{x^3-2},\ \ \ (-2,\ -\frac{1}{2})\) Text Transcription: f(x)=6/x^3 -2, (-2, -1/2)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(f(x)=\frac{1}{(x^2-3x)^2},\ \ \ (4,\ \frac{1}{16})\) Text Transcription: f(x)=1/(x^2 -3x)^2, (4, 1/16)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(f(t)=\frac{3t+2}{t-1},\ \ \ (0,\ -2)\) Text Transcription: f(t)=3t+2/t-1, (0, -2)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(f(x)=\frac{x+1}{2x-3},\ \ \ (2,\ 3)\) Text Transcription: f(x)=x+1/2x-3, (2, 3)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(y=26-\sec^3 4x,\ \ \ (0,\ 25)\) Text Transcription: y=26-sec^3 4x, (0, 25)

In Exercises 67–74, evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. \(y=\frac{1}{x}+\sqrt{\cos x},\ \ \ (\frac{\pi}{2},\ \frac{2}{\pi})\) Text Transcription: y=1/x + sqrtcos x, (pi/2, 2/pi)

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(f(x)=\sqrt{2x^2-7}\) (4, 5) Text Transcription: f(x)=sqrt2x^2 -7

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(f(x)=\frac{1}{3}x\sqrt{x^2+5}\) (2, 2) Text Transcription: f(x)=1/3 xsqrtx^2 +5

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(y=(4x^3+3)^2\) (-1, 1) Text Transcription: y=(4x^3 +3)^2

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(f(x)=(9-x^2)^{2/3}\) (1, 4) Text Transcription: f(x)=(9-x^2)^2/3

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(f(x)=\sin 2x\) \((\pi,\ 0)\) Text Transcription: f(x)=sin 2x (pi, 0)

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(y=\cos 3x\) \((\frac{\pi}{4},\ -\frac{\sqrt{2}}{2})\) Text Transcription: y = cos 3x (pi/4, -sqrt2/2)

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(f(x)=\tan^2 x\) \((\frac{\pi}{4},\ 1)\) Text Transcription: f(x)=tan^2 x (pi/4, 1)

In Exercises 75–82, (a) find an equation of the tangent line to the graph of f 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 the graphing utility to confirm your results. Function Point \(y=2 \tan^3 x\) \((\frac{\pi}{4},\ 2)\) Text Transcription: y=2 tan^3 x (pi/4, 2)

In Exercises 83–86, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(g(t)=\frac{3t^2}{\sqrt{t^2+2t-1}},\ \ \ (\frac{1}{2},\ \frac{3}{2})\) Text Transcription: g(t) = 3t^2 /sqrt t^2 +2t-1, (1/2 , 3/2)

In Exercises 83–86, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(f(x)=\sqrt{x}(2-x)^2,\ \ \ (4,\ 8)\) Text Transcription: f(x)=sqrtx (2-x)^2 , (4, 8)

In Exercises 83–86, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(s(t)=\frac{(4-2t)\sqrt{1+t}}{3},\ \ \ (0,\ \frac{4}{3})\) Text Transcription: s(t)=(4-2t)sqrt1+t /3, (0, 4/3)

In Exercises 83–86, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(y=(t^2-9)\sqrt{t+2},\ \ \ (2,\ -10)\) Text Transcription: y=(t^2 -9) sqrtt+2, (2, -10)

Famous Curves In Exercises 87 and 88, find an equation of the tangent line to the graph at the given point. Then use a graphing utility to graph the function and its tangent line in the same viewing window. Top half of circle

Famous Curves In Exercises 87 and 88, find an equation of the tangent line to the graph at the given point. Then use a graphing utility to graph the function and its tangent line in the same viewing window. Bullet-nose curve

Horizontal Tangent Line Determine the point(s) in the interval \((0,\ 2\pi)\) at which the graph of f(x) = 2 cos x + sin 2x has a horizontal tangent. Text Transcription: (0, 2pi)

Horizontal Tangent Line Determine the point(s) at which the graph of \(f(x)=\frac{x}{\sqrt{2x-1}}\) has a horizontal tangent. Text Transcription: f(x)=x/sqrt2x-1

In Exercises 91–96, find the second derivative of the function. \(f(x)=5(2-7x)^4\) Text Transcription: f(x)=5(2-7x)^4

In Exercises 91–96, find the second derivative of the function. \(f(x)=4(x^2-2)^3\) Text Transcription: f(x)=4(x^2 -2)^3

In Exercises 91–96, find the second derivative of the function. \(f(x)=\frac{1}{x-6}\) Text Transcription: f(x)=1/x-6

In Exercises 91–96, find the second derivative of the function. \(f(x)=\frac{4}{(x+2)^3}\) Text Transcription: f(x)=4/(x+2)^3

In Exercises 91–96, find the second derivative of the function. \(f(x)=\sin x^2\) Text Transcription: f(x)=sin x^2

In Exercises 91–96, find the second derivative of the function. \(f(x)=\sec^2 \pi\ x\) Text Transcription: f(x)=sec^2 pi x

In Exercises 97–100, evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(h(x)=\frac{1}{9}(3x+1)^3,\ \ \ (1,\ \frac{64}{9})\) Text Transcription: h(x)=1/9 (3x+1)^3 , (1, 64/9)

In Exercises 97–100, evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(f(x)=\frac{1}{\sqrt{x+4}},\ \ \ (0,\ \frac{1}{2})\) Text Transcription: f(x)=1/sqrtx+4 , (0, 1/2 )

In Exercises 97–100, evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(f(x)=\cos x^2,\ \ \ (0,\ 1)\) Text Transcription: f(x)=cos x^2 , (0, 1)

In Exercises 97–100, evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(g(t)=\tan 2t,\ \ \ (\frac{\pi}{6},\ \sqrt{3})\) Text Transcription: g(t) = tan 2t, (pi/6, sqrt3)

In Exercises 101-104, the graphs of a function f and its derivative f’ are shown. Label the graphs as f or f’ and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 101-104, the graphs of a function f and its derivative f’ are shown. Label the graphs as f or f’ and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 101-104, the graphs of a function f and its derivative f’ are shown. Label the graphs as f or f’ and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 101-104, the graphs of a function f and its derivative f’ are shown. Label the graphs as f or f’ and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 105 and 106, the relationship between f and g is given. Explain the relationship between f’ and g’. g(x) = f(3x)

In Exercises 105 and 106, the relationship between f and g is given. Explain the relationship between f’ and g’. \(g(x) = f(x^2)\) Text Transcription: g(x)=f(x^2)

Think About It The table shows some values of the derivative of an unknown function f. Complete the table by finding (if possible) the derivative of each transformation of f. (a) g(x) = f(x) - 2 (b) h(x) = 2 f(x) (c) r(x) = f (-3x) (d) s(x) = f(x + 2)

Given that g(5) = -3, g’(5) = 6, h(5) = 3, and h’(5) = -2, find f’(5) (if possible) for each of the following. If it is not possible, state what additional information is required. (a) \(f(x)=g(x)h(x)\) (b) \(f(x)=g(h(x))\) (c) \(f(x)=\frac{g(x)}{h(x)}\) (d) \(f(x)=[g(x)]^3\) Text Transcription: f(x)=g(x)h(x) f(x)=g(h(x)) f(x)=g(x)/h(x) f(x)=[g(x)]^3

In Exercises 109 and 110, the graphs of f and g are shown. Let h(x) = f(g(x)) and s(x) = g(f(x)). Find each derivative, if it exists. If the derivative does not exist, explain why. (a) Find h’(1). (b) Find s’(5)

In Exercises 109 and 110, the graphs of f and g are shown. Let h(x) = f(g(x)) and s(x) = g(f(x)). Find each derivative, if it exists. If the derivative does not exist, explain why. (a) Find h’(3). (b) Find s’(9).

Doppler Effect The frequency F of a fire truck siren heard by a stationary observer is \(F=132,400/(331 \pm v)\), where \(\pm v\) represents the velocity of the accelerating fire truck in meters per second (see figure). Find the rate of change of F with respect to v when (a) the fire truck is approaching at a velocity of 30 meters per second (use -v). (b) the fire truck is moving away at a velocity of 30 meters per second (use +v). \(F=\frac{132,400}{331 + v}\) \(F=\frac{132,400}{331 - v}\) Text Transcription: F=132,400/(331 pm v) pm v F=132,400/331 + v F=132,400/331 - v

Harmonic Motion The displacement from equilibrium of an object in harmonic motion on the end of a spring is \(y=\frac{1}{3}\cos 12t-\frac{1}{4}\sin 12t\) where y is measured in feet and t is the time in seconds. Determine the position and velocity of the object when \(t=\pi/8\). Text Transcription: y=1/3 cos 12t -1/4 sin 12t t=pi/8

Pendulum A 15-centimeter pendulum moves according to equation \(\theta = 0.2 \cos 8t\), where \(\theta\) is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement and the rate of change of \(\theta\) when t = 3 seconds. Text Transcription: theta = 0.2 cos 8t theta

Wave Motion A buoy oscillates in simple harmonic motion \(y=A \cos \omega t\) as waves move past it. The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds. (a) Write an equation describing the motion of the buoy if it is at its high point at t = 0. (b) Determine the velocity of the buoy as a function of t. Text Transcription: y = A cos omegat

Circulatory System The speed S of blood that is r centimeters from the center of an artery is \(S=C(R^2-r^2)\) where C is a constant, R is a radius of the artery, and S is measured in centimeters per second. Suppose a drug is administered and the artery begins to dilate at a rate of dR/dt. At a constant distance r, find the rate at which S changes with respect to t for \(C=1.76 \times 10^5\), \(R=1.2 \times 10^{-2}\), and \(dR/dt=10^{-5}\). Text Transcription: S=C(R^2 - r^2) C=1.76 times 10^5 R=1.2 times 10^-2 dR/dt = 10^-5

Modeling Data The normal daily maximum temperatures T (in degrees Fahrenheit) for Chicago, Illinois are shown in the table. (Source: National Oceanic and Atmospheric Administration) (a) Use a graphing utility to plot the data and find a model for the data of the form T(t) = a + b sin(ct - d) where T is the temperature and t is the time in months, with t = 1 corresponding to January. (b) Use a graphing utility to graph the model. How well does the model fit the data? (c) Find T’ and use a graphing utility to graph the derivative. (d) Based on the graph of the derivative, during what times does the temperature change most rapidly? Most slowly? Do your answers agree with your observations of the temperature changes? Explain.

Modeling Data The cost of producing x units of a product is C = 60x + 1350. For one week management determined the number of units produced at the end of t hours during an eight-hour shift. The average values of x for this week are shown in the table. (a) Use a graphing utility to fit a cubic model to the data. (b) Use the Chain Rule to find dC/dt. (c) Explain why the cost function is not increasing at a constant rate during the eight-hour shift.

Finding a Pattern Consider the function \(f(x)=\sin \beta x\), where \(\beta\) is a constant. (a) Find the first-, second-, third-, and fourth-order derivatives of the function. (b) Verify that the function and its second derivative satisfy the equation \(f''(x)+\beta^2f(x)=0\). (c) Use the results of part (a) to write general rules for the even- and odd-order derivatives \(f^{(2k)}(x)\ \text{and}\ f^{(2k-1)}(x)\). [Hint: \((-1)^k\) is positive if k is even and negative if k is odd.] Text Transcription: f(x)=sin beta x beta f’’(x) + beta^2 f(x) = 0 f^(2k) (x) and f^(2k-1) (x) (-1)^k

Conjecture Let f be a differentiable function of period p. (a) Is the function f’ periodic? Verify your answer. (b) Consider the function g(x) = f(2x). Is the function g’(x) periodic? Verify your answer.

Think About It Let r(x) = f(g(x)) and s(x) = g(f(x)), where f and g are shown in the figure. Find (a) r’(1) and (b) s’(4).

(a) Find the derivative of the function \(g(x)=\sin^2x+\cos^2x\) in two ways. (b) For \(f(x)=\sec^2x\ \text{and}\ g(x)=\tan^2x\), show that f’(x) = g’(x). Text Transcription: g(x)=sin^2 x + cos^2 x f(x)=sec^2 x and g(x)=tan^2 x

(a) Show that the derivative of an odd function is even. That is, if f(-x) = -f(x), then f’(-x) = f’(x). (b) Show that the derivative of an even function is odd. That is, if f(-x) = f(x), then f’(-x) = -f’(x).

Let u be a differentiable function of x. Use the fact that \(|u|=\sqrt{u^2}\) to prove that \(\frac{d}{dx}\ [|u|]=u'\frac{u}{|u|},\ \ \ u\ \neq\ 0\). Text Transcription: |u|=sqrtu^2 d/dx [|u|] = u’ u/|u|, u neq 0

In Exercises 124-127, use the result of Exercise 123 to find the derivative of the function. g(x) = |3x - 5|

In Exercises 124-127, use the result of Exercise 123 to find the derivative of the function. \(f(x)=|x^2-9|\) Text Transcription: f(x)=|x^2 -9|

In Exercises 124-127, use the result of Exercise 123 to find the derivative of the function. h(x) = |x| cos x

In Exercises 124-127, use the result of Exercise 123 to find the derivative of the function. f(x) = | sin x |

Linear and Quadratic Approximations The linear and quadratic approximations of a function f at x = a are \(P_1(x)=f'(a)(x-a)+f(a)\) and \(P_2(x)=\frac{1}{2}f''(a)(x-a)^2+f'(a)(x-a)+f(a)\). In Exercises 128 and 129, (a) find the specified linear and quadratic approximations of f, (b) use a graphing utility to graph f and the approximations, (c) determine whether \(P_1\) or \(P_2\) is the better approximation, and (d) state how the accuracy changes a you move farther from x = a. \(f(x)=\tan x\) \(a=\frac{\pi}{4}\) Text Transcription: P_1 (x) = f’(a)(x-a)+f(a) P_2 (x) = 1/2 f’’(a)(x-a)^2 + f’(a)(x-a)+f(a) P_1 P_2 f(x)=tan x a=pi/4

Linear and Quadratic Approximations The linear and quadratic approximations of a function f at x = a are \(P_1(x)=f'(a)(x-a)+f(a)\) and \(P_2(x)=\frac{1}{2}f''(a)(x-a)^2+f'(a)(x-a)+f(a)\). In Exercises 128 and 129, (a) find the specified linear and quadratic approximations of f, (b) use a graphing utility to graph f and the approximations, (c) determine whether \(P_1\) or \(P_2\) is the better approximation, and (d) state how the accuracy changes a you move farther from x = a. \(f(x)=\sec x\) \(a=\frac{\pi}{6}\) Text Transcription: P_1 (x) f’(a)(x-a)+f(a) P_2 (x) = 1/2 f’’(a)(x-a)^2 + f’(a)(x-a)+f(a) P_1 P_2 f(x)=sec x a=pi/6

True or False? In Exercises 130-132, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=(1-x)^{1/2}\), then \(y'=\frac{1}{2}(1-x)^{-1/2}\). Text Transcription: y=(1-x)^1/2 y’=1/2 (1-x)^-1/2

True or False? In Exercises 130-132, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=\sin^2(2x)\), then \(f'(x)=2(\sin 2x)(\cos 2x)\). Text Transcription: f(x)=sin^2 (2x) f’(x)=2(sin 2x)(cos 2x)

True or False? In Exercises 130-132, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y is a differentiable function of u, u is a differentiable function of v, and v is a differentiable function of x, then \(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dx}\). Text Transcription: dy/dx = dy/du du/dv dv/dx

Let \(f(x)=a_1sinx+a_2 sin 2x+ \cdot \cdot \cdot+a_n \sin nx\), where \(a_1,\ a_2,\ .\ .\ .\ ,\ a_n\) are real numbers and where n is a positive integer. Given that \(|f(x)|\ \leq\ |\sin x|\) for all real x, prove that \(|a_1+2a_2+ \cdot \cdot \cdot+na_n|\ \leq\ 1\). Text Transcription: f(x)=a_1 sin x + a_2 sin 2x + . . . + a_n sin nx a_1 , a_2 , . . . , a_n |f(x)| leq |sin x| |a_1 + 2a_2 + . . . + na_n| leq 1

Let k be a fixed positive integer. The nth derivative of \(\frac{1}{x^k-1}\) has the form \(\frac{P_n(x)}{(x^k-1)^{n+1}}\) where \(P_n(x)\) is a polynomial. Find \(P_n(1)\). Text Transcription: 1/x^k -1 P_n (x)/ (x^k -1)^n+1 P_n (x) P_n (1)