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\(\lim_{x\rightarrow0}{\frac{x^5+x^3}{5x^5+2x^3}}\) is
What are all horizontal asymptotes of the graph of \(y=\frac{7-e^x}{1+e^x}\) in the \(xy\)-plane?
For which of the following pairs of functions \(f\) and \(g\) is \(\lim_{x\rightarrow\infty}{\frac{f(x)}{g(x)}}\) infinite?
If \(y=\cos{\left(3x\right)}\), then \(\frac{dy}{dx}=\)
If \(f\left(x\right)=e^{\left(\frac{2}{x^2}\right)}\), then \(f^\prime\left(x\right)=\)
If \(f\left(x\right)=x^{10}+10x\), then \(\frac{d}{dx}\left(f\left(\sin{x}\right)\right)=\)
If \(y=\left(\frac{x}{5x-2}\right)^3\), then \(\frac{dy}{dx}=\)
\( f\left(x\right)=\left\{\begin{array}{ll} ax+b,\quad if\ {x}<{3} \\ x^2-ax, \quad if\ x\geq3\end{array}\right.\)
Let \(f\) be the function defined above, where \(a\) and \(b\) are constants. If \(f\) is differentiable at \(x=3\), what is the value of \(a+b\)?
Let \(f\) be the function defined by \(f\left(x\right)=x^3+x^2+2x\). Let \(g\left(x\right)=f^{-1}(x)\), where \(g\left(16\right)=2\). What is the value of \(g^\prime(16)\)?
\(\lim_{h\rightarrow0}{\frac{e^{-4-h}-e^{-4}}{h}}\)
If \(\sin{\left(xy\right)}=x^2\), then \(\frac{dy}{dx}=\)
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?
If \(f\left(x\right)=\cos\left(4x\right)\), then \(f^\prime\left(\frac{\pi}{8}\right)\)
The first derivative of the function \(f\) is given by \(f^\prime\left(x\right)=5x^4-20x^3\). What are the \(x\)-coordinates of the points of inflection of the graph of \(f\)?
Let \(f\) be a function with a second derivative given by \(f^{\prime\prime}\left(x\right)=x^6{(x-4)}^2(x-5)\). What are the \(x\)-coordinates of the points of inflection of the graph of \(f\)?
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?
A particle moves along a straight line. The graph of the particles position \(x(t)\) at time \(t\) is show above for \(0<t<4\). The graph has horizontal tangents at \(t=1\) and \(t=3\) and a point of inflection at \(t=2\). For what values of t is the velocity of the particle increasing?
The graph of the function \(f\) is shown in the figure above. Which of the following could be the graph of \(f^\prime\), the derivative of \(f\)?
The function \(f\) is twice differentiable with \(f\left(3\right)=2\), \(f^\prime\left(3\right)=6\), and \(f^{\prime\prime}\left(3\right)=1\). What is the value of the approximation of \(f\left(3.1\right)\) using the line tangent to the graph of \(f\) at \(x=3\)?
\(t\) (hours) |
0 | 3 | 7 | 10 |
---|---|---|---|---|
\(n(t)\) (gallons per hour) |
9 | 5 | 4 | 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 10 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 \(t=10\)?
\(\int\left(e^{3x}+2\pi\right)dx\)
\(\int{\sin{\left(x\right)}+\cos{\left(x\right)}dx}=\)
\(\int{{3x}^4\left(x^5-7\right)^5}dx\)
Shown above is a slope field for which of the following differential equations?
Which of the following is the solution to the differential equation \(\frac{dy}{dx}=\frac{10x^9y}{x^{10}+1}\) whose graph contains the point \((0, 1)\)?
If \(0<c<1\), what is the area of the region enclosed by the graphs of \(y=0\), \(y=\frac{5}{x}\), \(x=1\), and \(x=c\)?
A particle moves along the x-axis with the velocity given by \(v\left(t\right)=6t^5+8t^3\) for time \(t\geq0\). If the particle is at position \(x=1\) at time \(t=0\), what is the position of the particle at time \(t=1\)?
The graph of the function \(f\) shown above has horizontal tangent lines at \(x=1\) and \(x=4.5\). Let \(g\) be the function defined by \(g\left(x\right)=\int_{0}^{x}{f(t)}dt\). For what values of \(x\) does the graph of \(g\) have a point of inflection?
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 \(a\left(t\right)=90\sqrt[4]{t}\). The position of the particle is \(5\) when \(t=0\), and the position of the particle is \(39\) when \(t=1\). What is the velocity of the particle at \(t=0\)?