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Let \(f\) be a function that is continuous on the closed interval \([0, 2]\) with \(f\left(0\right)=7\) and \(f\left(2\right)=16\). Which of the following statements must be true?
The figure above shows the graph of a function \(f\) with domain \(-2\le x\le2\). Which of the following statements about \(f\) are true?
I. \({lim}_{x\rightarrow0^-}{f(x)}\) exists.
II. \({lim}_{x\rightarrow0^+}{f(x)}\) exists.
III. \({lim}_{x\rightarrow0}{f(x)}\) does not exist.
The graphs of \(f\) and \(g\) are shown above. If \(h\left(x\right)=f\left(x\right)g(x)\), then \(h^\prime\left(6\right)=\)
The radius of a sphere is decreasing at a rate of 4 centimeters per second. At the instant when the radius of the sphere is 3 centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface area \(S\) of a sphere with radius \(r\) is \(S=4\pi r^2\).)
The first derivative of the function \(g\) is given by \(g^\prime\left(x\right)=cos\left(-\frac{\pi}{2}x^2\right)\) for \(-.5<x<1.5\). On which of the following intervals is \(g\) decreasing?
The graph of the derivative of the function \(f\) is show in the figure above. The graph has horizontal tangent lines at \(x=-2\), \(x=-0.25\), and \(x=3\). At which of the following values of \(x\) does \(f\) have a relative minimum?
If \(\int_{-3}^{2}f\left(x\right)dx=-7\) and \(\int_{6}^{2}f\left(x\right)dx=-2\), what is the value of \(\int_{-3}^{6}{f(x)}dx\)?
\(x\) | 2 | 3 | 4 | 5 |
---|---|---|---|---|
\(f(x)\) | 1.25 | -1.75 | -3.25 | 3.5 |
\(f^\prime(x)\) | -8 | 0 | 1.75 | 4.5 |
The table above gives values of a function \(f\) and its derivative at selected values of \(x\). If \(f^\prime\) is continuous on the interval \([2,5]\), what is the value of \(\int_{5}^{2}{f^\prime\left(x\right)}dx\)?
The graph of the function \(f\), which as a domain of \([0, 5]\), is shown in the figure above. The graph consists of a quarter circle of radius \(2\) and a segment with slope \(1\). Let \(b\) be a positive number such that \(\int_{b}^{0}{f(x)}dx=0\). What is the value of \(b\)?
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 \((-1, 0)\) and \(h\left(x\right)=5x\bullet f(x)\), then \(h^\prime\left(-1\right)=\)
A forest located beside a grassland has rectangular boundary as shown in the figure above. The eagle density of the forest at any point along a strip \(x\) kilometers from the grassland’s edge is \(f(x)\) eagles per square kilometer. Which of the following expressions gives the eagle population of the forest?
What is the area enclosed by the curves \(y={x}^3-{8x}^2+13x+2\) and \(y=x+3\)?
What is the average value of \(y=\frac{\sin(x)}{3x^2-2x-2}\) on the closed interval \([2, 5]\)?
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?