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True or False If a function \(f\) is defined and continuous on a closed interval \([a,b],\) differentiable on the open interval \((a,b),\) and if \(f(a)=f(b)\), then Rolle’s Theorem guarantees that there is at least one number \(c\) in the interval \(( a,b)\) for which \(f( c) =0.\)

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In your own words, give a geometric interpretation of the Mean Value Theorem.

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True or False If two functions \(f\) and \(g\) are differentiable on an open interval \(( a,b)\) and if \(f^\prime (x) =g^\prime (x)\) for all numbers \(x\) in \(( a,b)\), then \(f\) and \(g\) differ by a constant on \((a, b)\).

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True or False When the derivative \(f^\prime\) is positive on an open interval \(I\), then \(f\) is positive on \(I\).

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

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

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\(g(x)=x^{2}-2x-2\) on \([0,2]\)

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\(g(x)=x^{2}+1\) on \([-1,1]\)

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

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

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

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

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\(s(t)=t^{4}-2t^{2}+1\) on \([-2,2]\)

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\(s(t)=t^{4}+t^{2}\) on \([-2,2]\)

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\(f(x)=\sin (2x)\) on \([0,\pi ]\)

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\(f(x)=\sin x+\cos x\) on \([0,2\pi ]\)

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

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

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

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\(f(x)=x^{2/5}\) on \([-1,1]\)

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

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\(f(x)=x+2+\dfrac{3}{x-1}\) on \([2,7]\)

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\(f(x)=\ln \sqrt{x}\) on \([1,e]\)

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\(f(x)=xe^{x}\) on \([0,1]\)

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\(f(x)=x^{3}-5x^{2}+4x-2\) on \([1,3]\)

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\(f(x)=x^{3}-7x^{2}+5x\) on \([-2,2]\)

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

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

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\(f(x)=\sqrt[3]{x^{2}}\) on \([1,8]\)

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

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\(f(x)=x^{3}+6x^{2}+12x+1\)

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\(f(x)=-x^{3}+3x^{2}+4\)

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\(f(x)=x^{2/3}(x^{2}-4)\)

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\(f(x)=x^{1/3}(x^{2}-7)\)

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\(f(x)=\vert x^{3}+3 \vert\)

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\(f(x)=|{x^{2}-4}|\)

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\(f(x)=3\sin x\) on \([0,2\pi]\)

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\(f(x)=\cos (2x)\) on \([0,2\pi]\)

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\(f(x) =xe^{x}\)

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\(g(x) =x+e^{x}\)

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\(f(x) =e^{x}\sin x\), \(0\leq x\leq 2\pi\)

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\(f(x) =e^{x}\cos x\), \(0\leq x\leq 2\pi\)

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Show that the function \(f(x)=2x^{3}-6x^{2}+6x-5\) is increasing for all \(x.\)

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Show that the function \(f(x)=x^{3}-3x^{2}+3x\) is increasing for all \(x.\)

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Show that the function \(f(x)=\dfrac{x}{x+1}\) is increasing on any interval not containing \(x=-1\).

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Show that the function \(f(x)=\dfrac{x+1}{x}\) is decreasing on any interval not containing \(x=0\).

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Mean Value Theorem Draw the graph of a function \(f\) that is continuous on \([a,b]\) but not differentiable on \(( a,b) ,\) and for which the conclusion of the Mean Value Theorem does not hold.

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Mean Value Theorem Draw the graph of a function \(f\) that is differentiable on \((a,b)\) but not continuous on \([a,b],\) and for which the conclusion of the Mean Value Theorem does not hold.

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Rectilinear Motion An automobile travels \(20\) mi down a straight road at an average velocity of \(40\) mph. Show that the automobile must have a velocity of exactly \(40\) mph at some time during the trip. (Assume that the distance function is differentiable.)

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Rectilinear Motion Suppose a car is traveling on a highway. At \(4:00\, {\rm p.m.}\), the car’s speedometer reads \(40\) mph. At \(4:12 \,{\rm p.m.}\), it reads \(60\) mph. Show that at some time between \(4:00\) and \(4:12 {\rm p.m.}\), the acceleration was exactly \(100\,{\rm{mi}}/{{\rm{h}}^2}\).

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Rectilinear Motion Two stock cars start a race at the same time and finish in a tie. If \(f_{1}(t)\) is the position of one car at time \(t\) and \(f_{2}(t)\) is the position of the second car at time \(t\), show that at some time during the race they have the same velocity. (Hint: Set \(f(t)=f_{2}(t)-f_{1}(t)\).)

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Rectilinear Motion Suppose \(s=f(t)\) is the distance \(s\) that an object has traveled from the origin at time \(t\). If the object is at a specific location at \(t=a,\) and returns to that location at \(t=b,\) then \(f(a)=f(b)\). Show that there is at least one time \(t=c,\) \(a<c<b\) for which \(f^\prime ( c) =0\). That is, show that there is a time \(c\) when the velocity of the object is \(0\).

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Loaded Beam The vertical deflection \(d\) (in feet), of a particular \(5\)-\({\rm foot}\)-long loaded beam can be approximated by \[ d=d(x) =-\dfrac{1}{192}x^{4}+\dfrac{25}{384}x^{3}-\dfrac{25}{128}x^{2} \]

where \(x\) (in feet) is the distance from one end of the beam.

1. Verify that the function \(d=d(x)\) satisfies the conditions of Rolle’s Theorem on the interval \([0,5]\)
2. What does the result in (a) say about the ends of the beam?
3. Find all numbers \(c\) in \((0,5)\) that satisfy the conclusion of Rolle’s Theorem. Then find the deflection \(d\) at each number \(c.\)
4. Graph the function \(d\) on the interval \([0,5]\).

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For the function \(f(x)=x^{4}-2x^{3}-4x^{2}+7x+3\):

1. Find the critical numbers of \(f\) rounded to three decimal places.
2. Find the intervals where \(f\) is increasing and decreasing.

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Rolle’s Theorem Use Rolle’s Theorem with the function \(f(x)=( x-1) \sin x\) on \([0,1]\) to show that the equation \(\tan x+x=1\) has a solution in the interval \((0,1)\).

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Rolle’s Theorem Use Rolle’s Theorem to show that the function \(f(x) =x^{3}-2\) has exactly one real zero.

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Rolle’s Theorem Use Rolle’s Theorem to show that the function \(f(x) =(x-8) ^{3}\) has exactly one real zero.

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Rolle’s Theorem Without finding the derivative, show that if \(f(x)=(x^{2}-4x+3)(x^{2}+x+1)\), then \(f^\prime (x)=0\) for at least one number between \(1\) and \(3\). Check by finding the derivative and using the Intermediate Value Theorem.

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Rolle’s Theorem Consider \(f(x)=|x|\) on the interval \([-1,1]\). Here, \(f(1)=\,f(-1)=1\) but there is no \(c\) in the interval \((-1,1)\) at which \(f^\prime (c)=0\). Explain why this does not contradict Rolle’s Theorem.

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Mean Value Theorem Consider \(f(x)=x^{2/3}\) on the interval \([-1,1]\). Verify that there is no \(c\) in \(({-}1,1)\) for which \[ f^\prime (c)=\frac{f(1)-f(-1)}{1-(-1)} \]

Explain why this does not contradict the Mean Value Theorem.

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Mean Value Theorem The Mean Value Theorem guarantees that there is a real number \(N\) in the interval \((0,1)\) for which \(f^\prime (N)=f(1)-f(0)\) if \(f\) is continuous on the interval \([0,1]\) and differentiable on the interval \((0,1)\). Find \(N\) if \(f(x)= \sin ^{-1}x\).

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Mean Value Theorem Show that when the Mean Value Theorem is applied to the function \(f(x)=Ax^{2}+Bx+C\) in the interval \([a,b]\), the number \(c\) referred to in the theorem is the midpoint of the interval.

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1. Apply the Increasing/Decreasing Function Test to the function \(f(x) =\sqrt{x}.\) What do you conclude?
2. Is \(f\) increasing on the interval \([0, \infty)\)? Explain.

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Explain why the function \(f(x)=ax^{4}+bx^{3}\,+cx^{2}+dx\,+e\,\) must have a zero between \(0\) and \(1\) if \[ \frac{a}{5}+\frac{b}{4}+\frac{c}{3}+\frac{d}{2}+e=0 \]

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Put It Together If \(f^\prime (x)\) and \(g^\prime (x)\) exist and \(f^\prime (x)>g^\prime (x)\) for all real \(x\), then which of the following statements must be true about the graph of \(y=f(x)\) and the graph of \(y=g(x)\)?

1. They intersect exactly once.
2. They intersect no more than once.
3. They do not intersect.
4. They could intersect more than once.
5. They have a common tangent at each point of intersection.

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Prove that there is no \(k\) for which the function \[ f(x)=x^{3}-3x+k \]

has two distinct zeros in the interval \([0,1]\).

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Show that \(e^{x}>x^{2}\) for all \(x>0\).

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Show that \(e^{x}>1+x\) for all \(x>0\). (Hint: Show that \(f(x)=e^{x}-1-x\) is an increasing function for \(x>0\).)

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Show that \(0<\ln x<x\) for \(x>1\).

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Show that \(\tan \theta \geq \theta\) for all \(\theta\) in the open interval \(\left( 0,\dfrac{\pi }{2}\right).\)

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Let \(f\) be a function that is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\). If \(f(x)=0\) for three different numbers \(x\) in \((a,b),\) show that there must be at least two numbers in \((a,b)\) at which \(f^\prime (x) =0\).

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Proof for the Increasing/Decreasing Function Test Let \(f\) be a function that is continuous on a closed interval \([ a,b]\). Show that if \(f^\prime (x) <0\) for all numbers in \(( a,b)\), then \(f\) is a decreasing function on \(( a,b)\). (See the Corollary on p. 279.)

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Suppose that the domain of \(f\) is an open interval \(( a,b)\) and \(f^\prime (x)>0\) for all \(x\) in the interval. Show that \(f\) cannot have an extreme value on \((a,b)\).

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Use Rolle’s Theorem to show that between any two real zeros of a polynomial function \(f\), there is a real zero of its derivative function \(f^\prime\).

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Find where the general cubic \(f(x)=ax^{3}+bx^{2}+cx+d\) is increasing and where it is decreasing by considering cases depending on the value of \(b^{2}-3ac\). (Hint: \(f^\prime (x)\) is a quadratic function; examine its discriminant.)

284

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Explain why the function \(f(x) =\) \(x^{n}+ax+b,\) where \(n\) is a positive even integer, has at most two distinct real zeros.

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Explain why the function \(f(x) =\) \(x^{n} + ax + b,\) where \(n\) is a positive odd integer, has at most three distinct real zeros.

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Explain why the function \(f(x) =\) \(x^{n} + ax^{2} + b,\) where \(n\) is a positive odd integer, has at most three distinct real zeros.

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Explain why the function \(f(x) =\) \(x^{n} + ax^{2} + b,\) where \(n\) is a positive even integer, has at most four distinct real zeros.

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Mean Value Theorem Use the Mean Value Theorem to verify that \[ \frac{1}{9}<\sqrt{66}-8<\frac{1}{8} \]

(Hint: Consider \(f(x)=\sqrt{x}\) on the interval \([64,66]\).)

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Given \(f(x)=\dfrac{ax^{n}+b}{cx^{n}+d}\), where \(n\geq 2\) is a positive integer and \(ad-bc\neq 0\), find the critical numbers and the intervals on which \(f\) is increasing and decreasing.

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Show that \(x\leq \ln (1+x)^{1+x}\) for all \(x>-1\). (Hint: Consider \(f(x)=-x+\ln (1+x)^{1+x}\).)

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Show that \(a\ln \dfrac{b}{a}\leq b-a<b\ln \dfrac{b}{a}\) if \(0<a<b\).

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Show that for any positive integer \(n,\,\frac{n}{{n + 1}} <\ln {\left( {1 + \frac{1}{n}} \right)^n} < 1.\)