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If \({lim}_{x\rightarrow a^-}{f(x)}=5,{lim}_{x\rightarrow a^+}{f(x)}=5,\ and\ f(a)=3\), then which of the following statements about \(f\) must be true?
The vertical line \(x=3\) is an asymptote for the graph of the function \(f\). Which of the following statements must be false?
\(x\) | 0 | 3 | 5 | 7 | 8 | 10 |
---|---|---|---|---|---|---|
\(f(x)\) | \(15\) | 6 | 8 | 11 | 0 | \(-15\) |
Let \(f\) be a differentiable function with selected values given in the table above. What is the average rate of change of \(f\) over the closed interval \(0\le x\le10\)?
\(x\) | \(f(x)\) | \(f^\prime(x)\) | \(g(x)\) | \(g^\prime(x)\) |
---|---|---|---|---|
1 | 5 | \(7\) | 8 | \(3\pi\) |
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 \(y=f(x)\) define a twice-differentiable function and let \(y=r(x)\) be the line tangent to the graph of \(f\) at \(x=3\). If \(r(x)\le f(x)\) for all real \(x\), which of the following must be true?
The function \(P(t)\) models the population of elk in Rocky Mountain National Park, in thousands of elk, where \(t\) is the number of years since January 1, 2020. Which of the following is the best interpretation of the statement \(P^\prime(2)=3.682\)?
The first derivative of the function \(f\) is given by \(f^\prime(x)=sin\left(x^2\right)\). At which of the following values of \(x\) does \(f\) have a local maximum?
A particle moves along the \(x\)-axis so that its position at time \(t>0\) is given by \(x(t)\) and \(\frac{dx}{dt}=-15t^4+7t^2+3t\). The acceleration of the particle is zero when \(t=\)
The figure above shows the graph of \(f’\), the derivative of a function \(f\), for \(0≤x≤6\). What is the value of \(x\) at which the absolute minimum of \(f\) occurs?
The function \(f\) is continuous on the closed interval \([1,8]\). If \(\int_{1}^{8}{f(x)dx}=23\) and \(\int_{8}^{3}{f(x)dx}=-15\), then \(\int_{1}^{3}{2f(x)dx}=\)
If \(f^\prime(x)=cos\left(x^2\right)\) and \(f(3)=3\), then \(f(2)=\)
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?
The rate at which honey is leaking from a beehive is modeled by the function \(B\) defined by \(B(t)=2+sin\left(t^2\right)\) 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 half hour?
What is the area of the region enclosed by the graphs of \(y=\sqrt{16x-x^4}\) and \(y=\frac{x}{2}\)?
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 \(t=11\) minutes?