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\(\lim_{x\rightarrow0}{\frac{6x^{13}+3x^4}{7x^{13}+9x^4}}\) is
\(\lim_{x\rightarrow\infty}{\frac{\left(3x1\right)\left(27x^2\right)}{\left(x2\right)\left(8x^2+4\right)}}=\)
If \(f\left(x\right)=x^{4}+4x\), then \(\frac{d}{dx}\left(f\left(\cos{x}\right)\right)=\)
If \(f\left(x\right)=e^{\left(\frac{8}{x}\right)}\), then \(f^\prime\left(x\right)=\)
If \(f\left(x\right)=x^5+5x\), then \(\frac{d}{dx}\left(f\left(\ln{x}\right)\right)=\)
If \(f(x)=(x8){(x^2+2)}^6\), then \(f^\prime\left(x\right)=\)
\( f\left(x\right)=\left\{\begin{array}{ll} \frac{x^236}{x6},\quad if\ x\neq6 \\ 12, \quad if\ x=6\end{array}\right.\)
Let \(f\) be the function defined above. Which of the following statements about \(f \) are true?
I. \(f \) has a limit at \( x=6\).
II. \(f \) is continuous at \( x=6\).
III. \(f \) is differentiable at \( x=6\).
Let \(f\) be the differentiable function such that \(f\left(4\right)=2,\ f\left(3\right)=4,\ f^\prime\left(4\right)=3,\ and\ f^\prime\left(3\right)=9\). The function \(g\) is differentiable and \(g\left(x\right)=f^{1}(x)\) for all \(x\). What is the value of \(g^\prime(4)\)?
\(\frac{d}{dx}\left({sin}^{1}x+4\sqrt x\right)=\)
If \(e^{xy}y^2=e9\), then at \(x=\frac{1}{3}\) and \(y=3\), \(\frac{dy}{dx}=\)
If \(\sin{\left(2xy\right)}=x\), then \(\frac{dy}{dx}=\)
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 \(f\) increasing?
What is the slope of the line tangent to the curve \(y=\arccos(2x)\) at the point at which \(x=\frac{1}{4}\)?
In the \(xy\)plane, the line \(x+y=k\), where \(k\) is a constant, is tangent to the graph of \(y=2x^2+3x+1\). What is the value of \(k\)?
\(x\)  1  2  3  4 

\(f^{\prime\prime}\left(x\right)\)  6  0  5  3 
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?
\(x\)  \(f(x)\) 

3  10 
0  3 
4  9 
8  12 
The table above gives selected values for a twicedifferentiable function \(f\). Which of the following must be true?
The points \((4, 0)\), \((x, 0)\), \(\left(x,\frac{1}{x^2}\right)\), and \(\left(4,\frac{1}{x^2}\right)\) are the vertices of a rectangle, where \(x\geq4\), as shown in the figure above. For what value of \(x\) does the rectangle have a maximum area?
The graph of the function \(f\) is shown above. Which of the following could be the graph of \(f^\prime\), the derivative of \(f\)?
The slope of the tangent line to the graph of \(y=\ln(12x)\) at \(x=3\) is
The graph of the function \(f\) is shown above for \(0\le\ x\le4\). Of the following, which has the least value?
What are all values of \(x\) for which \(\int_{x}^{2}t^4dt\) is equal to 0?
\(\int{\frac{1}{x^6}dx}=\)
\(\int\frac{3x^2}{x^36}dx\)
\(\int_{2}^{0}\frac{3x^2}{\sqrt{x^3+1}}dx\)
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 \(\frac{dy}{dx}=\frac{x^3}{y}\) 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 \(\int_{9}^{14}{f(t)}dt\) 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 \([3, 5]\)?
Let \(h\) be the function defined by \(h\left(x\right)=\int_{\frac{3\pi}{4}}^{x}{{cos}^2t}dt\). Which of the following is an equation for the line tangent to the graph of \(h\) at the point where \(x=\frac{3\pi}{4}\)?
If \(y=f(x)\) is a solution to the differential equation \(\frac{dy}{dx}=e^{x^3}\) with the initial condition \(f\left(0\right)=3\), which of the following is true?