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True or False A polynomial function is continuous at every real number.

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True or False Piecewise-defined functions are never continuous at numbers where the function changes equations.

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The three conditions necessary for a function \(f\) to be continuous at a number \(c\) are _______, _______, and _______.

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True or False If \(f(x)\) is continuous at \(0\), then \(g(x)=\dfrac{1}{4}f(x)\) is continuous at \(0\).

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True or False If \(f\) is a function defined everywhere in an open interval containing \(c,\) except possibly at \(c\), then the number \(c\) is called a removable discontinuity of \(f\) if the function \(f\) is not continuous at \(c\).

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True or False If a function \(f\) is discontinuous at a number \(c,\) then \(\lim\limits_{x\rightarrow c}f( x)\) does not exist.

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True or False If a function \(f\) is continuous on an open interval \((a,b),\) then it is continuous on the closed interval \([a,b]\).

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True or False If a function \(f\) is continuous on the closed interval \([a,b]\), then \(f\) is continuous on the open interval \((a,b)\).

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The velocity of a ball thrown up into the air as a function of time, if the ball lands \(5\) seconds after it is thrown and stops.

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The temperature of an oven used to bake a potato as a function of time.

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True or False If a function \(f\) is continuous on a closed interval \([ a,b]\), then the Intermediate Value Theorem guarantees that the function takes on every value between \(f(a)\) and \(f(b)\).

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True or False If a function \(f\) is continuous on a closed interval \([a, b]\) and \(f( a) \neq f( b)\), but both \(f( a) >0\) and \(f( b) >0,\) then according to the Intermediate Value Theorem, \(f\) does not have a zero on the open interval \(( a,b).\)

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\(c=-3\)

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\(c=0\)

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\(c=2\)

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\(c=3\)

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\(c=4\)

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\(c=5\)

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\({f}( x) = x^2+1 \) at \(c=-1\)

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\({f}( x) = x^3-5 \) at \(c=5\)

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\(f(x)=\dfrac{x}{x^{2}+4}\) at \(c=-2\)

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\(f(x)=\dfrac{x}{x-2}\) at \(c=2\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} 2x+5 & \hbox{if }x\leq 2 \\[3pt] 4x+1 & \hbox{if }x>2 \end{array} \right. \) at \(c=2\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} 2x+1 & \hbox{if }x\leq 0 \\[3pt] 2x & \hbox{if }x>0 \end{array} \right.\) at \(c=0\)

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\(f(x)=\left\{ \begin{array}{c@{\quad}l} 3x-1 & \hbox{if }x\lt 1 \\[3pt] 4 & \hbox{if }x=1 \\[3pt] 2x & \hbox{if }x>1 \end{array} \right. \) at \(c=1\)

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\(f(x)=\left\{ \begin{array}{c@{\quad}l@{\quad}l} 3x-1 & \hbox{if }x\lt 1 \\[3pt] 2 & \hbox{if }x=1 & \hbox{at}\ c=1\\[3pt] 2x & \hbox{if }x>1 \end{array} \right.\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} 3x-1 & \hbox{if }x\lt 1 \\[3pt] 2x & \hbox{if }x>1 \end{array} \right.\) at \(c=1\)

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\(f(x)=\left\{ \begin{array}{c@{\quad}l} 3x-1 & \hbox{if }x\lt 1 \\[3pt] 2 & \hbox{if }x=1 \\[3pt] 3x & \hbox{if }x>1 \end{array} \right.\) at \(c=1\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} x^{2} & \hbox{if }x\leq 0 \\[3pt] 2x & \hbox{if }x>0 \end{array} \right.\) at \(c=0\)

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\(f(x)=\left\{ \begin{array}{c@{\quad}l} x^{2} & \hbox{if }x\lt -1 \\[3pt] 2 & \hbox{if }x=-1 \\[3pt] -3x+2 & \hbox{if }x>-1 \end{array} \right.\) at \(c=-1\)

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\(f(x)=\left\{ \begin{array}{c@{\quad}l} 4-3x^{2} & \hbox{if }x\lt 0 \\ 4 & \hbox{if }x=0 \\[3pt] \sqrt{\dfrac{16-x^{2}}{4-x}} & \hbox{if }0\lt x\lt 4 \end{array} \right.\)  at \(c=0\)

103

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\(f(x)=\left\{ \! \begin{array}{l@{\ \ }l} \sqrt{4+x} & \hbox{if \(-4\)}\leq x\leq 4 \\[3pt] \sqrt{\dfrac{x^{2}-3x-4}{x-4}} & \hbox{if } x>4 \end{array} \right.\) at \(c=4\)

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\({f}( x) =\dfrac{x^{2}-4}{x-2}\), \(c=2\)

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\({f}( x) =\dfrac{x^{2}+x-12}{x-3}\), \(c=3\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} 1+x & \hbox{if }x \lt 1 \\[3pt] 4 & \hbox{if } x=1\\[3pt] 2x & \hbox{if }x>1 \end{array} \right.\)  \(c=1\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} x^{2}+5x & \hbox{if }x\lt -1 \\ 0 & \hbox{if } x=-1\\[3pt] x-3 & \hbox{if }x>-1 \end{array} \right.\) \(c=-1\)

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\({f}( x) =\dfrac{x^{2}-9}{x-3}\) on the interval \([ -3,3)\)

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\({f}( x) =1+\dfrac{1}{x}\) on the interval \([-1,0)\)

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\({f}( x) =\dfrac{1}{\sqrt{x^{2}-9}}\) on the interval \([ -3,3]\)

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\({f}( x) =\sqrt{9-x^{2}}\) on the interval \([ -3,3]\)

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\(f( x) =2x^{2}+5x-\dfrac{1}{x}\)

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\(f( x) =x+1+\dfrac{2x}{x^{2}+5}\)

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\(f( x) =( x-1) (x^{2}+x+1) \)

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\(f( x) =\sqrt{x} (x^{3}-5)\)

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\(f(x)=\dfrac{x-9}{\sqrt{x}-3}\)

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\(f(x)={\dfrac{{x-4}}{{\sqrt{x}-2}}}\)

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\(f(x)=\sqrt{{\dfrac{{x^{2}+1}}{{2-x}}}}\)

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\(f(x)=\sqrt{{\dfrac{{4}}{{x^{2}-1}}}}\)

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\(f( x) = (2x^{2}+5x-3) ^{2/3}\)

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\(f( x) =( x+2) ^{1/2}\)

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Is \(f\) continuous at \(0\)? Why or why not?

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Is \(f\) continuous at \(4\)? Why or why not?

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Is \(f\) continuous at \(3\)? Why or why not?

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Is \(f\) continuous at \(2\)? Why or why not?

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Is \(f\) continuous at \(1\)? Why or why not?

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Is \(f\) continuous at \(2.5\)? Why or why not?

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\(f( x) =\dfrac{x^{3}-8}{x-2}\)

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\(f( x) =\dfrac{x^{2}-3x+2}{3x-6}\)

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\(f(x)=x^{3}-3x\) on \({[{-2,2}]}\)

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\(f(x)=x^{4}-1\) on \({[{-2,2}]}\)

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\(f(x)={\dfrac{{x}}{{({x+1})^{2}}}}-1\) on \([ 10,20]\)

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\(f(x)=x^{3}-2x^{2}-x+2\) on \({[{3,4}]\qquad }\)

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\(f( x) =\dfrac{x^{3}-1}{x-1}\) on \([ 0,2]\)

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\(f(x)=\dfrac{x^{2}+3x+2}{ x^{2}-1}\) on \({[-{3,0}]}\)

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\(f( x) = x^{3}+3x-5\); interval: \(( 1,2)\)

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\(f( x) =x^{3}-4x+2\); interval: \((1 ,2)\)

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\(f( x) =2x^{3}+3x^{2}+4x-1\); interval: \((0,1)\)

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\(f( x) =x^{3}-x^{2}-2x+1\); interval: \((0,1)\)

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\(f( x) =x^{3}-6x-12\); interval:\((3,4)\)

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\(f( x) =3x^{3}+5x-40\); interval: \((2,3)\)

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\(f( x) =x^{4}-2x^{3}+21x-23\); interval: \(( 1,2)\)

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\(f( x) =x^{4}-x^{3}+x-2\); interval: \(( 1,2)\)

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\(f( x) =\sqrt{x^{2}+4x}-2\) in \((0, 1)\)

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\(f(x)=x^{3}-x+2\) in \((-2, 0)\)

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\(u_{1}( t) =\left\{ \begin{array}{l@{\quad}l} 0 & \hbox{if }t\lt 1 \\[3pt] 1 & \hbox{if }t\geq 1 \end{array} \right.\quad c=1\)

104

Question

\(u_{3}( t) =\left\{ \begin{array}{l@{\quad}l} 0 & \hbox{if }t\lt 3 \\ 1 & \hbox{if }t \geq 3 \end{array} \quad c=3\right.\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} 1-x^{2} & \hbox{if }\vert x\vert \leq 1 \\ x^{2}-1 & \hbox{if }\vert x\vert >1 \end{array} \right.\)

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\(f(x)=\left\{ \begin{array}{c@{\quad}c} \sqrt{4-x^{2}} & \hbox{if }\vert x\vert \leq 2 \\ \left\vert \,x\,\right\vert -2 & \hbox{if }\vert x\vert >2 \end{array} \right.\)

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First-Class Mail As of January 2013, the U.S. Postal Service charged $0.46 postage for first-class letters weighing up to and including 1 ounce, plus a flat fee of $0.20 for each additional or partial ounce up to 3.5 ounces. First-class letter rates do not apply to letters weighing more than 3.5 ounces.

1. Find a function \(C\) that models the first-class postage charged for a letter weighing \(w\) ounces. Assume \(w>0\).
2. What is the domain of \(C?\)
3. Determine the intervals on which \(C\) is continuous.
4. At numbers where \(C\) is not continuous, what type of discontinuity does \(C\) have?
5. What are the practical implications of the answer to (d)?

Source: U.S. Postal Service Notice 123.

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First-Class Mail As of January 2013, the U.S. Postal Service charged $0.92 postage for first-class retail flats (large envelopes) weighing up to and including 1 ounce, plus a flat fee of $0.20 for each additional or partial ounce up to 13 ounces. First-class rates do not apply to flats weighing more than 13 ounces.

1. Find a function \(C\) that models the first-class postage charged for a large envelope weighing \(w\) ounces. Assume \(w>0\).
2. What is the domain of \(C?\)
3. Determine the intervals on which \(C\) is continuous.
4. At numbers where \(C\) is not continuous (if any), what type of discontinuity does \(C\) have?
5. What are the practical implications of the answer to (d)?

Source: U.S. Postal Service Notice 123.

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Cost of Natural Gas In February 2012 Peoples Energy had the following monthly rate schedule for natural gas usage in single- and two-family residences with a single gas meter:

Source: Peoples Energy, Chicago, IL.

1. Find a function \(C\) that models the monthly cost of using \(x\) therms of natural gas.
2. What is the domain of \(C?\)
3. Determine the intervals on which \(C\) is continuous.
4. At numbers where \(C\) is not continuous (if any), what type of discontinuity does \(C\) have?
5. What are the practical implications of the answer to (d)?

Question

Cost of Water The Jericho Water District determines quarterly water costs, in dollars, using the following rate schedule:

Source: Jericho Water District, Syosset, NY.

1. Find a function \(C\) that models the quarterly cost of using \(x\) thousand gallons of water.
2. What is the domain of \(C\)?
3. Determine the intervals on which \(C\) is continuous.
4. At numbers where \(C\) is not continuous (if any), what type of discontinuity does \(C\) have?
5. What are the practical implications of the answer to (d)?

Question

Gravity on Europa Europa, one of the larger satellites of Jupiter, has an icy surface and appears to have oceans beneath the ice. This makes it a candidate for possible extraterrestrial life. Because Europa is much smaller than most planets, its gravity is weaker. If we think of Europa as a sphere with uniform internal density, then inside the sphere, the gravitational field \(g\) is given by \(g( r) =\dfrac{Gm}{R^{3}}r\), \(0\leq r\lt R\), where \(R\) is the radius of the sphere, \(r\) is the distance from the center of the sphere, and \(G\) is the universal gravitation constant. Outside a uniform sphere of mass \(m\), the gravitational field \(g\) is given by \[ g( r) =\dfrac{Gm}{r^{2}}, R\lt r. \]

1. For the gravitational field of Europa to be continuous at its surface, what must \(g(r)\) equal? [Hint: Investigate \(\lim\limits_{r\rightarrow R}g( r).]\)
2. Determine the gravitational field at Europa's surface. This will indicate the type of gravity environment organisms will experience. Use the following measured values: Europa's mass is \(4.8\times 10^{22}\)kilograms, its radius is \(1.569 \times 10^6\) meters, and \(G=6.67\times 10^{-11}\).
3. Compare the result found in (b) to the gravitational field on Earth's surface, which is 9.8 meter/second\(^2\). Is the gravity on Europa less than or greater than that on Earth?

105

Question

Find constants \(A\) and \(B\) so that the function below is continuous for all \(x\). Graph the resulting function. \begin{equation*} f(x)=\left\{ \begin{array}{c@{\qquad}lrll} ( x-1) ^{2} & \hbox{if } &- \infty&\lt x\lt 0 \\ \left( A-x\right) ^{2} & \hbox{if }&0&\leq x\lt 1 \\ x+B & \hbox{if }&1&\leq x\lt \infty \end{array} \right. \end{equation*}

Question

Find constants \(A\) and \(B\) so that the function below is continuous for all \(x\). Graph the resulting function.\begin{equation*} f(x)=\left\{ \begin{array}{c@{\qquad}lrll} x+A & \hbox{if }&-\infty &\lt x\lt 4 \\ ( x-1) ^{2} & \hbox{if }&4&\leq x\leq 9 \\ Bx+1 & \hbox{if }&9&\lt x\lt \infty \end{array} \right. \end{equation*}

Question

For the function \(f\) below, find \(k\) so that \(f\) is continuous at \(2\). \begin{equation*} f(x)=\left\{ \begin{array}{c@{\qquad}l} \dfrac{\sqrt{2x+5}-\sqrt{x+7}}{x-2}, &x\geq -\dfrac{5}{2}, x\neq 2 \\ k & \hbox{if }\ x=2\end{array} \right. \end{equation*}

Question

Suppose \(f(x) ={\dfrac{{x^{2}-6x-16}}{{( {x^{2}-7x-8}) \sqrt{x^{2}-4}}}}\).

1. For what numbers \(x\) is \(f\) defined?
2. For what numbers \(x\) is \(f\) discontinuous?
3. Which discontinuities found in (b) are removable?

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Intermediate Value Theorem

1. Use the Intermediate Value Theorem to show that the function \(f(x)=\sin x+x-3\) has a zero in the interval \([0,\pi]\).
2. Approximate the zero rounded to three decimal places.

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Intermediate Value Theorem

1. Use the Intermediate Value Theorem to show that the function \(f(x)=e^{x}+x-2\) has a zero in the interval \([0,2]\).
2. Approximate the zero rounded to three decimal places.

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Graph a function that is continuous on the closed interval \([5,12]\), that is negative at both endpoints and has exactly three zeros in this interval. Does this contradict the Intermediate Value Theorem? Explain.

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Graph a function that is continuous on the closed interval \([-1,2]\), that is positive at both endpoints and has exactly two zeros in this interval. Does this contradict the Intermediate Value Theorem? Explain.

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Graph a function that is continuous on the closed interval \([-2,3]\), is positive at \(-2\) and negative at \(3\) and has exactly two zeros in this interval. Is this possible? Does this contradict the Intermediate Value Theorem? Explain.

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Graph a function that is continuous on the closed interval \([-5,0]\), is negative at \(-5\) and positive at \(0\) and has exactly three zeros in the interval. Is this possible? Does this contradict the Intermediate Value Theorem? Explain.

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1. Explain why the Intermediate Value Theorem gives no information about the zeros of the function \(f(x)=x^{4}-1\) on the interval \([-2,2].\)
2. Use technology to determine whether or not \(f\) has a zero on the interval \([-2, 2]\).

Question

1. Explain why the Intermediate Value Theorem gives no information about the zeros of the function \(f(x)=\) \(\ln ( x^{2}+2)\) on the interval \([-2,2].\)
2. Use a graphing technology to determine whether or not \(f\) has a zero on the interval \([-2, 2]\).

Question

Intermediate Value Theorem

1. Use the Intermediate Value Theorem to show that the functions \(y=x^{3}\) and \(y=1-x^{2}\) intersect somewhere between \(x=0\) and \(x=1.\)
2. Use graphing technology to find the coordinates of the point of intersection rounded to three decimal places.
3. Use graphing technology to graph both functions on the same set of axes. Be sure the graph shows the point of intersection.

Question

Intermediate Value Theorem An airplane is travelling at a speed of 620 miles per hour and then encounters a slight headwind that slows it to 608 miles per hour. After a few minutes, the headwind eases and the plane's speed increases to 614 miles per hour. Explain why the plane's speed is 610 miles per hour on at least two different occasions during the flight.

Source: Submitted by the students of Millikin University.

Question

Suppose a function \(f\) is defined and continuous on the closed interval \([a,b]\). Is the function \(h(x)=\dfrac{1}{f( x) }\) also continuous on the closed interval \([a,b]\)? Discuss the continuity of \(h\) on \([a, b]\).

Question

Given the two functions \(f\) and \(h\): \begin{equation*} {f}( x) =x^{3}-3x^{2}-4x+12\qquad h( x) =\left\{ \begin{array}{c@{\quad}l} \dfrac{f( x) }{x-3} & \hbox{if }x\neq 3 \\[5pt] p & \hbox{if }x=3 \end{array} \right. \end{equation*}

1. Find all the zeros of the function \(f\).
2. Find the number \(P\) so that the function \(h\) is continuous at \(x=3\). Justify your answer.
3. Determine whether \(h\), with the number found in (b), is even, odd, or neither. Justify your answer.

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The function \(f(x)=\dfrac{|x|}{x}\) is not defined at \(0\). Explain why it is impossible to define \(f(0)\) so that \(f\) is continuous at \(0\).

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Find two functions \(f\) and \(g\) that are each continuous at \(c\), yet \(\dfrac{f}{g}\) is not continuous at \(c\).

Question

Discuss the difference between a discontinuity that is removable and one that is nonremovable. Give an example of each.

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\(f(x) =x^{3}+3x-5\); interval: \((1,2)\)

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\(f(x) =\) \(x^{3}-4x+2\); interval: \((1, 2)\)

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\(f(x) =2x^{3}+3x^{2}+4x-1\); interval: \((0,1)\)

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\(f( x)=x^{3}-x^{2}-2x+1\); interval: \((0,1)\)

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\(f( x) =x^{3}-6x-12\); interval: \((3,4)\)

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\(f( x)=3x^{3}+5x-40\); interval: \((2,3)\)

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\(f( x) =x^{4}-2x^{3}+21x-23\); interval \((1,2)\)

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\(f( x)=x^{4}-x^{3}+x-2\); interval: \((1,2)\)

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Intermediate Value Theorem Use the Intermediate Value Theorem to show that the function \(f( x) =\sqrt{x^{2}+4x}-2\) has a zero in the interval \([0,1]\). Then approximate the zero correct to one decimal place.

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Intermediate Value Theorem Use the Intermediate Value Theorem to show that the function \(f(x)=x^{3}-x+2\) has a zero in the interval \([-2,0]\). Then approximate the zero correct to two decimal places.

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Continuity of a Sum If \(f\) and \(g\) are each continuous at \(c\), prove that \(f+g\) is continuous at \(c\). [Hint: Use the Limit of a Sum Property.)

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Intermediate Value Theorem Suppose that the functions \(f\) and \(g\) are continuous on the interval \([a,b].\) If \(f(a)\lt g(a)\) and \(f(b)>g(b),\) prove that the graphs of \(y=f(x)\) and \(y=g(x)\) intersect somewhere between \(x=a\) and \(x=b\). [Hint: Define \(h(x) = f( x) -g(x)\) and show \(h(x) =0\) for some \(x\) between \(a\) and \(b\).]

Question

Intermediate Value Theorem Let \(f(x)=\dfrac{1}{x-1}+ \dfrac{1}{x-2}.\) Use the Intermediate Value Theorem to prove that there is a real number \(c\) between \(1\) and \(2\) for which \(f(c)=0\).

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Intermediate Value Theorem Prove that there is a real number \(c\) between \(2.64\) and \(2.65\) for which \(c^{2}=7\).

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Show that the existence of \({\lim\limits_{h\rightarrow 0}} \dfrac{f( a+h) -f( a) }{h}\) implies \(f(x)\) is continuous at \(x=a\).

Question

Find constants \(A,B,C\), and \(D\) so that the function below is continuous for all \(x\). Sketch the graph of the resulting function. \begin{equation*} f(x)=\left\{ \begin{array}{l@{\qquad}lrll} \dfrac{x^{2}+x-2}{x-1} & \hbox{if} & -\infty &\lt x\lt 1 \\ A & \hbox{if } & x&=1 \\ B\left( x-C\right) ^{2} & \hbox{if} & 1&\lt x\lt 4 \\ D & \hbox{if} & x&=4 \\ 2x-8 & \hbox{if} & 4&\lt x \lt \infty \end{array} \right. \end{equation*}