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A function \(F\) is called a(n) _____________________ of a function \(f\) if \(F^\prime =f.\)

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True or False If \(F\) is an antiderivative of \(f\), then \(F(x) +C\), where \(C\) is a constant, is also an antiderivative of \(f\).

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All the antiderivatives of \(y=x^{-1}\) are _____________________.

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True or False An antiderivative of \(\sin x\) is \(-\cos x+\pi.\)

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True or False The general solution of a differential equation \(\dfrac{\textit{dy}}{\textit{dx}}=f(x)\) consists of all the antiderivatives of \(f^\prime (x)\).

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True or False Free fall is an example of motion with constant acceleration.

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True or False To find a particular solution of a differential equation \(\dfrac{\textit{dy}}{\textit{dx}}=f(x) ,\) we need a boundary condition.

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True or False If \(F_{1}\) and \(F_{2}\) are both antiderivatives of a function \(f\) on an interval \(I\), then \(F_{1}-F_{2}=C,\) a constant on \(I\).

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\(f(x) =2\)

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\(f(x) =\dfrac{1}{2}\)

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\(f(x)=4x^{5}\)

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\(f(x)=x\)

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

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

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

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

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\(f(x)=\sqrt{x}\)

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\(f(x)=\dfrac{1}{\sqrt{x}}\)

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\(f(x)=4x^{3}-3x^{2}+1\)

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

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

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

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

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\(f(x)=\dfrac{4x^{3/2}-1}{x}\)

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\(f(x)=2x-3\;\cos\;x\)

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\(f(x)=2\;\sin\;x-\;\cos\;x\)

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\(f(x) =4e^{x}+x\)

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

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\(f(x) =\dfrac{7}{1+x^{2}}\)

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\(f(x) =x+\dfrac{10}{\sqrt{1-x^{2}}}\)

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\(\dfrac{\textit{dy}}{\textit{dx}}=3x^{2}-2x+1,\quad {\rm when}\, x=2,\, y=1\)

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\(\dfrac{dv }{dt}=3t^{2}-2t+1,\quad {\rm when}\, t=1,\, v=5\)

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\(\dfrac{\textit{dy}}{\textit{dx}}=x^{1/3}+x\sqrt{x}-2,\quad {\rm when}\, x=1,\, y=2\)

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\(s^\prime (t) =t^{4}+4t^{3}-5,\quad s( 2) =5\)

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\(\dfrac{\textit{ds}}{\textit{dt}}=t^{3}+\dfrac{1}{t^{2}}, \quad {\rm when}\, t=1,\, s=2\)

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\(\dfrac{\textit{dy}}{\textit{dx}}=\sqrt{x}-x\sqrt{x}+1, \quad {\rm when}\, x=1,\, y=0\)

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\(f^\prime (x) =x-2\;\sin\;x,\quad f( \pi ) =0 \)

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\(\dfrac{\textit{dy}}{\textit{dx}}=x^{2}-2\;\sin\;x, \quad {\rm when}\, x=\pi ,\, y=0\)

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\(\dfrac{d^{2}y}{dx^{2}}=e^{x},\, {\rm when}\, x=0,\, y=2, \quad {\rm when}\;x=1,\, y=e\)

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\(f^{\prime \prime} (\theta ) =\sin \theta +\cos \theta ,\, f^{\prime}\,\left( \dfrac{\pi }{2}\right) =2\, {\rm and}\, f( \pi ) =4\)

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\(a=-32\;{{\rm ft}}/{{\rm s}}^{2}, \quad s(0)=0\;{{\rm ft}},\quad v(0)=128\;{{\rm ft}}/{{\rm s}}\)

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\(a=-980\;{{\rm cm}}/{{\rm s}}^{2}, \quad s(0)=5\;{{\rm cm}}, \quad v(0)=1980\;{{\rm cm/s}}\)

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\(a=3t\;{{\rm m}}/{{\rm s}}^{2}, \quad s(0)=2\;{{\rm m}}, \quad v(0)=18\;{{\rm m}}/{{\rm s}}\)

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\(a=5t-2\;{{\rm ft}}/{{\rm s}}^{2}, \quad s(0)=0\;{{\rm ft}}, \quad v(0)=8\;{{\rm ft}}/{{\rm s}}\)

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\(f(u)=\dfrac{u^{2}+10u+21}{3u+9}\)

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\(f(t)=\dfrac{t^{3}-5t+8}{t^{5}}\)

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\(f^\prime (t) =\dfrac{t^{4}+3t-1}{t}\) if \(f( 1) =\dfrac{1}{4}\)

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\(g^\prime (x) =\dfrac{x^{2}-1}{x^{4}-1}\) if \(g( 0)=0\)

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Use the fact that \[ \frac{d}{\textit{dx}}({x\;\cos\;x+\sin\;x})=-x\;\sin\;x+2\;\cos\;x \]

to find \(F\) if \[ {\frac{\textit{dF}}{\textit{dx}}=-x\;\sin\;x+2\;\cos\;x}\qquad \hbox{and}\qquad {F(0)=1} \]

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Use the fact that \[ \frac{d}{\textit{dx}}\;\sin\;x^{2}=2x\;\cos\;x^{2} \]

to find \(h\) if \[ {\frac{\textit{dh}}{\textit{dx}}=x\;\cos\;x^{2}}\qquad \hbox{and}\qquad {h(0)=2} \]

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Rectilinear Motion A car decelerates at a constant rate of \(10\;{\rm m}/{\rm s}^{2}\) when its brakes are applied. If the car must stop within 15 m after applying the brakes, what is the maximum allowable velocity for the car? Express the answer in \({\rm m}/{\rm s}\) and in \({\rm mi/h}.\)

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Rectilinear Motion A car can accelerate from \(0\) to \(60\;{\rm km}/{\rm h}\) in \(10\) seconds. If the acceleration is constant, how far does the car travel during this time?

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Rectilinear Motion A BMW 6 series can accelerate from \(0\) to \(60\) \({\rm mph}\) in \(5\) seconds. If the acceleration is constant, how far does the car travel during this time?

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Free Fall The \(2\)-m high jump is common today. If this event were held on the Moon, where the acceleration due to gravity is \(1.6\;{\rm m}/{\rm s}^{2}\), what height would be attained? Assume that an athlete can propel him- or herself with the same force on the Moon as on Earth.

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Free Fall The world’s high jump record, set on July 27, 1993, by Cuban jumper Javier Sotomayor, is \(2.45\;{\rm m}\). If this event were held on the Moon, where the acceleration due to gravity is \(1.6\;{\rm m}/{\rm s}^{2}\), what height would Sotomayor have attained? Assume that he propels himself with the same force on the Moon as on Earth.

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Free Fall A ball is thrown straight up from ground level, with an initial velocity of \(19.6\;{\rm m}/{\rm s}\). How high is the ball thrown? How long will it take the ball to return to ground level?

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Free Fall A child throws a ball straight up. If the ball is to reach a height of \(9.8\;{\rm m}\), what is the minimum initial velocity that must be imparted to the ball? Assume the initial height of the ball is \(1\;{\rm m}\).

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Free Fall A ball thrown directly down from a roof \(49 \,{\rm m}\) high reaches the ground in \(3\) seconds. What is the initial velocity of the ball?

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Inertia A constant force is applied to an object that is initially at rest. If the mass of the object is 4 \({\rm kg}\) and if its velocity after \(6\) seconds is \(12\;{\rm m}/{\rm s}\), determine the force applied to it.

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Rectilinear Motion Starting from rest, with what constant acceleration must a car move to travel \(2\) km in \(2\) min? (Give your answer in centimeters per second squared.)

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Downhill Speed of a Skier The down slope acceleration \(a\) of a skier is given by \(a=a(t) =g\;\sin\;\theta\), where \(t\) is time, in seconds, \(g=9\). \(8\;{\rm m}/{\rm s}^{2}\) is the acceleration due to gravity, and \(\theta\) is the angle of the slope. If the skier starts from rest at the lift, points his skis straight down a \(20^{\circ}\) slope, and does not turn, how fast is he going after 5 seconds?

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Free Fall A child on top of a building \(24 \,{\rm m}\) high drops a rock and then \(1\) second later throws another rock straight down. What initial velocity must the second rock be given so that the dropped rock and the thrown rock hit the ground at the same time?

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Radiation Radiation, such as X-rays or the radiation from radioactivity, is absorbed as it passes through tissue or any other material. The rate of change in the intensity \(I\) of the radiation with respect to the depth \(x\) of tissue is directly proportional to the intensity \(I\). This proportion can be expressed as an equation by introducing a positive constant of proportionality \(k\), where \(k\) depends on the properties of the tissue and the type of radiation.

1. Show that \(\dfrac{dI}{dx}=-kI\), \(k>0.\)
2. Explain why the minus sign is necessary.
3. Solve the differential equation in (a) to find the intensity \(I\) as a function of the depth \(x\) in the tissue. The intensity of the radiation when it enters the tissue is \(I( 0) =I_{0}\).
4. Find the value of \(k\) if the intensity is reduced by \(90\%\) of its maximum value at a depth of \(2.0\;{\rm cm}\).

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Moving Shadows A lamp on a post \(10\;{\rm m}\) high stands 25 m from a wall. A boy standing \(5\;{\rm m}\) from the lamp and \(20\;{\rm m}\) from the wall throws a ball straight up with an initial velocity of \(19.6\;{\rm m}/{\rm s}\). The acceleration due to gravity is \(a=-9.8\;{\rm m}/{\rm s}^{2}\). The ball is thrown up from an initial height of \(1\;{\rm m}\) above ground.

1. How fast is the shadow of the ball moving on the wall \(3\) seconds after the ball is released?
2. Explain if the ball is moving up or down.
3. How far is the ball above ground at \(t=3\) seconds?