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Let \(f\) be a function that is continuous on the closed interval \([-1, 1]\) with \(f\left(-1\right)=3\) and \(f\left(1\right)=-3\). Which of the following statements must be true?
If \(f\) is a continuous function such that \(f\left(3\right)=10\), which of the following statements must be true?
The graph of the function \(f\) is shown above. Which of the following limits does not exist?
The graphs of \(f\) and \(g\) are shown above. If \(h\left(x\right)=f\left(x\right)g(x)\), then \(h^\prime\left(4\right)=\)
The function \(f\) is continuous on the closed interval \([4, 6]\) and twice differentiable on the open interval \((4, 6)\). If \(f^\prime\left(5\right)=1\) and \(f^{\prime\prime}(x)<0\) on the open interval \((4, 6)\), which of the following could be a table of values for \(f\)?
In the \(xy\)-plane, the graph of the twice-differentiable function \(y=f(x)\) is concave up on the open interval \((1, 3)\) and is tangent to the line \(y=5x-3\) at \(x=2\). Which of the following statements must be true about the derivative of \(f\)?
Let \(f\) be a twice-differentiable function on the open interval \((2, 7)\). If \(f^\prime\left(x\right)>0\) on \((a, b)\) and \(f^{\prime\prime}\left(x\right)<0\) on \((2, 7)\), which of the following could be the graph of \(f\)?
\(t\) | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|
\(v(t)\) | 15 | 2 | -5 | -6 | 0 |
The table gives selected values of the velocity, \(v(t)\), of a particle moving along the \(x\)-axis. At time \(t=0\), the particle is at the origin. Which of the following could be the graph of the position, \(x(t)\), of the particle for \(0\le t\le4\)?
The first derivative of the function \(f\) is defined by \(f^\prime(x)=cos\left(x-x^2\right)\) for \(-2\le x\le3\). On what intervals is \(f\) increasing?
If \(\int_{4}^{-7}f\left(x\right)dx=-13\) and \(\int_{4}^{7}f\left(x\right)dx=-5\), what is the value of \(\int_{-7}^{7}{f(x)}dx\)?
\(x\) | 1 | 2 | 3 | 4 |
---|---|---|---|---|
\(f(x)\) | 9 | 5 | 7 | 8 |
\(f^\prime(x)\) | -1 | 1 | 5 | 7 |
The derivative of the function \(f\) is continuous on the closed interval \([1, 5]\). Values of \(f\) and \(f^\prime(x)\) for selected values of \(x\) are given in the table above. If \(\int_{2}^{5}{f^\prime(t)}dt=0\), then \(f\left(5\right)=\)?
A slope field for a differential equation is shown in the figure above. If \(y=f(x)\) is particular solution to the differential equation through the point \((0, 1)\) and \(h\left(x\right)=5x\bullet f(x)\), then \(h^\prime\left(0\right)=\)
An electric tractor uses electricity at the rate \(g\left(t\right)=3+2cos\left(\frac{t}{100}\right)\) kilowatts per minute, where \(t\) is the number of minutes since starting the tractor. To the nearest kilowatt, what is the total amount of kilowatts used from \(t = 0\) to \(t = 30\) minutes?
What is the area enclosed by the curves \(y={x}^3-{8x}^2+13\) and \(y=x+9\)?
A particle moves along a straight line for 8 seconds so that its velocity, in meters per second, is modeled by the graph shown above. During the time interval \(0\le t\le8\), what is the total distance the particle travels?