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\( f\left(x\right)=\left\{\begin{array}{ll} x^2sin\left(\pi x\right),\quad for\ x<2 \\ x^2+cx-6, \quad for\ x\geq2\end{array}\right.\)
Let \(f\) be the function defined above, where \(c\) is a constant. For what value of \(c\), if any, is \(f\) continuous at \(x=2\)?
\(\lim_{x\rightarrow-\infty}{\frac{\sqrt{49x^4+5}}{16x^2+4}}\) is
If \(f(x)=x^5+7\) and \(g\) is a differentiable function of \(x\), what is the derivative of \(f(g(x))\)?
If \(y=4\cos{\left(9x\right)}\), then \(\frac{dy}{dx}=\)
If \(f\left(x\right)=x^{5}+5x\), then \(\frac{d}{dx}\left(f\left(\sin{x}\right)\right)=\)
\( f\left(x\right)=\left\{\begin{array}{ll} ax+b,\quad if\ {x}<{0} \\ x^2-ax, \quad if\ x\geq0\end{array}\right.\)
Let \(f\) be the function defined above, where \(a\) and \(b\) are constants. If \(f\) is differentiable at \(x=0\), what is the value of \(a-b\)?
Let \(f\) be the function defined by \(f\left(x\right)=x^3+x^2+3x\). Let \(g\left(x\right)=f^{-1}(x)\), where \(g\left(-3\right)=-1\). What is the value of \(g^\prime(-3)\)?
If \(y=sinx\ cosx\), then at \(x=\frac{\pi}{6}\), \(\frac{dy}{dx}=\)
If \(e^{xy}-y^2=e-36\), then at \(x=-\frac{1}{6}\) and \(y=-6\), \(\frac{dy}{dx}=\)
Dirt is deposited into a pile with circular base. The volume \(V\) of the pile is given by \(V=\frac{r^3}{3}\), where \(r\) is the radius of the base, in meters. The circumference of the base is increasing at a constant rate of \(4\pi\) meters per hour. When the circumference of the base is \(2\pi\) meters, what is the rate of change of the volume of the pile, in cubic meters per hour?
Let \(f\) be the function with derivative given by \(f^\prime\left(x\right)=\frac{5x^3}{\left(3+x^2\right)^2}\). On what interval is \(f\) decreasing?
\(f(x)=\sqrt x+\frac{3}{\sqrt x}\), then \(f^\prime(9)=\)
The velocity, \(v\), in meters per second, of a certain type of wave is given by \(v(h)=3\sqrt h\), where \(h\) is the depth, in meters, of the water through which the wave moves. What is the rate of change, in meters per second per meter, of the velocity of the wave with respect to the depth of the water, when the depth is 4 meters?
\(x\) | 2 | 3 | 4 | 5 |
---|---|---|---|---|
\(f^{\prime\prime}\left(x\right)\) | -3 | 2 | 0 | -7 |
The polynomial function \(f\) has selected values of its second derivative \(f^{\prime\prime}\) given in the table above. Which of the following statements must be true?
Let \(f\) be the function given by \(f\left(x\right)=x^3+2x\). For what value of \(x\) in the closed interval \([2, 5]\) does the instantaneous rate of change of \(f\) equal the average rate of change of \(f\) on that interval?
The graph of the function \(f\) is shown above. Which of the following could be the graph of \(f^\prime\), the derivative of \(f\)?
\(t\) (hours) |
0 | 3 | 5 | 8 |
---|---|---|---|---|
\(n(t)\) (gallons per hour) |
8 | 5 | 3 | 2 |
Nitrogen is flowing into a room at the rate of \(n(t)\), where \(n(t)\) is measured in gallons per hour and \(t\) is measured in hours. The tank contains 13 gallons of nitrogen at time \(t=0\). Values of \(n(t)\) for selected values of \(t\) are given in the table above. Using a trapezoidal sum with three intervals indicated by the table, what is the approximation of the number of gallons of nitrogen in the room at time \(t=8\)?
\(\int\left(2e^{2x}+3e\right)dx\)
Which of the following is an antiderivative of \(7\ {csc}^2x+9\)?
\(\int{{9x}^3\left(x^4-2\right)^9}dx\)
\(\int_{0}^{1}\frac{{8x}^3}{\sqrt[3]{7x^4+1}}dx\)
Avian flu spreads among a population of \(P\) chickens at a rate proportional to the product of the number of chickens who have the disease and the number of chickens who do not have the disease. If \(c\) denotes the number of chickens who had the disease, which of the following differential equations could be used to model the situation with respect to time \(t\), where \(k\) is a positive constant?
Which of the following is the solution to the differential equation \(\frac{dy}{dx}=x^3y\) with the initial condition \(y\left(0\right)=-2\)?
A function \(f(t)\) gives the rate at which oil is leaking into the ocean, in gallons per hour, from a leaking underwater pipe, where t is measured in hours since 12 midnight. Which of the following gives the meaning of \(\int_{13}^{16}{f(t)}dt\) in the context described?
Let \(f\) be the function defined by \(f\left(x\right)=\frac{1}{x}\). What is the average value of \(f\) on the interval \([5, 10]\)?
A particle moves along the x-axis with the velocity given by \(v\left(t\right)=15t^4+8t^3\) for time \(t\geq0\). If the particle is at position \(x=4\) at time \(t=0\), what is the position of the particle at time \(t=1\)?
Let \(h\) be the function defined by \(h\left(x\right)=\int_{2\pi}^{x}{{cos}^2t}dt\). Which of the following is an equation for the line tangent to the graph of \(h\) at the point where \(x=2\pi\)?
If \(y=f(x)\) is a solution to the differential equation \(\frac{dy}{dx}=\ln\left(x^4\right)\) with the initial condition \(f\left(1\right)=3\), which of the following is true?
The graph of the piecewise linear function \(f\) is shown in the figure above. If \(g\left(x\right)=\int_{-2}^{x}f\left(t\right)dt\), which one of the following values is greatest?
A particle moves along the \(x\)-axis so that at time \(t\geq0\), the acceleration of the particle is \(a\left(t\right)=91\sqrt[6]{t}\). The position of the particle is \(3\) when \(t=0\), and the position of the particle is \(45\) when \(t=1\). What is the velocity of the particle at \(t=0\)?