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\(\lim_{x\rightarrow0}{\frac{3x^5+7x^3}{5x^5+21x^3}}\) is
\(\lim_{x\rightarrow-\infty}{\frac{\left(3x-1\right)\left(2-7x^2\right)}{\left(x-2\right)\left(8x^2+4\right)}}=\)
\(\lim_{x\rightarrow\infty}{\frac{\sqrt{25x^4+5}}{3x^2-9}}\) is
For which of the following pairs of functions \(f\) and \(g\) is \(\lim_{x\rightarrow\infty}{\frac{f(x)}{g(x)}}\) infinite?
If \(f\left(x\right)=e^{\left(\frac{1}{x}\right)}\), then \(f^\prime\left(x\right)=\)
If \(f\left(x\right)=x^{12}+12x\), then \(\frac{d}{dx}\left(f\left(\ln{x}\right)\right)=\)
If \(f(x)=(x+8){(x^2+2)}^6\), then \(f^\prime\left(x\right)=\)
\( f\left(x\right)=\left\{\begin{array}{ll} -3ax+b,\quad if\ {x}<{-3} \\ x^2-ax, \quad if\ x\geq-3\end{array}\right.\)
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 differentiable function such that \(f\left(5\right)=2,\ f\left(7\right)=5,\ f^\prime\left(5\right)=-3,\ and\ f^\prime\left(7\right)=2\). The function \(g\) is differentiable and \(g\left(x\right)=f^{-1}(x)\) for all \(x\). What is the value of \(g^\prime(5)\)?
\(\lim_{h\rightarrow0}{\frac{e^{-9-h}-e^{-9}}{h}}\)
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 \({10}\pi\) 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?
In the \(xy\)-plane, the line \(4x+y=k\), where \(k\) is a constant, is tangent to the graph of \(y=3x^2+2x+2\). What is the value of \(k\)?
The first derivative of the function \(f\) is given by \(f^\prime\left(x\right)=5x^3+15x^2\). What are the \(x\)-coordinates of the points of inflection of the graph of \(f\)?
The points \((-1, 0)\), \((x, 0)\), \(\left(x,\frac{1}{x^2}\right)\), and \(\left(-1,\frac{1}{x^2}\right)\) are the vertices of a rectangle, where \(x\leq{-1}\), as shown in the figure above. For what value of \(x\) does the rectangle have a maximum area?
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 decreasing?
The slope of the tangent line to the graph of \(y=\ln(2-3x)\) at \(x=-3\) is
The function \(f\) is twice differentiable with \(f\left(4\right)=3\), \(f^\prime\left(4\right)=5\), and \(f^{\prime\prime}\left(4\right)=2\). What is the value of the approximation of \(f\left(4.1\right)\) using the line tangent to the graph of \(f\) at \(x=4\)?
What are all values of \(x\) for which \(\int_{x}^{1}t^3dt\) is equal to 0?
\(\int\left(e^{2x}+e\right)dx\)
\(\int\frac{x^4}{{5x}^5-7}dx\)
\(\int{x^2\left(x^3-8\right)^4}dx\)
Shown above is a slope field for which of the following differential equations?
Which of the following is the solution to the differential equation \(\frac{dy}{dx}=x^3y\) 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 \(\int_{2}^{9}{f(t)}dt\) in the context described?
If \(0<c<1\), what is the area of the region enclosed by the graphs of \(y=0\), \(y=\frac{2}{x}\), \(x=c\), and \(x=1\)?
Let \(h\) be the function defined by \(h\left(x\right)=\int_{\frac{5\pi}{4}}^{x}{{sin}^2t}dt\). Which of the following is an equation for the line tangent to the graph of \(h\) at the point where \(x=\frac{5\pi}{4}\)?
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?