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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?
The vertical line \(x=4\) is an asymptote for the graph of the function \(f\). Which of the following statements must be false?
The graph of the function \(h\) is shown in the figure above. Of the following, which has the greatest value?
\(x\) | \(f(x)\) | \(f^\prime(x)\) | \(g(x)\) | \(g^\prime(x)\) |
---|---|---|---|---|
1 | \(3\pi\) | \(7\) | \(-2\pi\) | 5 |
The table above gives values of the differentiable functions \(f\) and \(g\) and their derivatives at \(x=1\). If \(h(x)=\frac{f(x)}{g(x)}\), what is the value of \(h^\prime(1)\)?
Let \(f\) be a twice-differentiable function on the open interval \((3, 7)\). If \(f^\prime\left(x\right)<0\) on \((a, b)\) and \(f^{\prime\prime}\left(x\right)>0\) on \((3, 7)\), which of the following could be the graph of \(f\)?
The first derivative of the function \(g\) is given by \(g^\prime\left(x\right)=sin\left(\frac{\pi}{4}x^2\right)\) for \(-.5<x<1.7\). On which of the following intervals is \(g\) increasing?
A particle moves along a straight line with the velocity given by \(v\left(t\right)=5+e^\frac{t}{7}\) for time \(t\geq0\). What is the acceleration of the particle at time \(t = 1\)?
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 maximum?
The graph of \(f^\prime\), the derivative of \(f\), is show above. The line tangent to the graph of \(f^\prime\) at \(x=0\) is vertical, and\(f^\prime\) is not differentiable at \(x=2\). Which of the following statements is true?
The function \(f\) is continuous on the closed interval \([1,9]\). If \(\int_{1}^{9}{f(x)dx}=13\) and \(\int_{9}^{5}{f(x)dx}=36\), then \(\int_{1}^{5}{2f(x)dx}=\)
The graph of the piecewise linear function \(f\) is shown above. Let \(h\) be the function given by \(h(x)=\int_{-3}^{x}{f(t)}dt\). On which of the following intervals is \(h\) increasing?
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 \((0, 0)\) and \(h\left(x\right)=5x\bullet f(x)\), then \(h^\prime\left(0\right)=\)
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?
The rate at which honey is leaking from a beehive is modeled by the function \(B\) defined by \(B(t)=3+sin\left(t^2\right)\) for time \(t\geq0\). \(B(t)\) is measured in liters per hour, and \(t\) is measured in hours. How much honey leaks out of the beehive during the first 30 minutes?
The temperature \(F\), in degrees Fahrenheit (\(°F\)), of a cup of coffee \(t\) minutes after it is poured is given by \(F(t)=68+115e^{-0.083t}\). To the nearest degree, what is the average temperature of the coffee between \(t=0\) and \(t=10\) minutes?