#2 Sample Exam AP Calculus AB Multiple Choice (Untimed-No Calculator Allowed)

\(\lim_{x\rightarrow0}{\frac{x^5+x^3}{5x^5+2x^3}}\) is

What are all horizontal asymptotes of the graph of \(y=\frac{9-5^x}{3+5^x}\) in the \(xy\)-plane?

For which of the following pairs of functions \(f\) and \(g\) is \(\lim_{x\rightarrow\infty}{\frac{f(x)}{g(x)}}\) zero?

If \(y=2\cos{\left(8x\right)}\), then \(\frac{dy}{dx}=\)

If \(f\left(x\right)=e^{\left(\frac{8}{x^2}\right)}\), then \(f^\prime\left(x\right)=\)

If \(f\left(x\right)=x^{11}+11x\), then \(\frac{d}{dx}\left(f\left(\sin{x}\right)\right)=\)

If \(y=\left(\frac{2x}{7x+1}\right)^3\), then \(\frac{dy}{dx}=\)

\( f\left(x\right)=\left\{\begin{array}{ll} ax+b,\quad if\ {x}<{0} \\ x^2-ax, \quad if\ x\geq0\end{array}\right.\)

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 \(b-a\)?

Let \(f\) be the function defined by \(f\left(x\right)=x^3+2x^2+x\). Let \(g\left(x\right)=f^{-1}(x)\), where \(g\left(4\right)=1\). What is the value of \(g^\prime(4)\)?

\(\lim_{h\rightarrow0}{\frac{e^{-3-h}-e^{-3}}{h}}\)

If \(\sin{\left(xy\right)}=x^5\), 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 \({10}\pi\) 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 \(f^\prime\left(\frac{\pi}{2}\right)\)

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 \(f\)?

Let \(f\) be a function with a second derivative given by \(f^{\prime\prime}\left(x\right)=x^4(x-2)(x+4)\). 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 \([-5, -2]\) 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 \(0<t<3\). The graph has horizontal tangents at \(t=.5\) and \(t=2\) and a point of inflection at \(t=1\). For what values of t is the velocity of the particle decreasing?

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 \(f\left(4\right)=3\), \(f^\prime\left(4\right)=5\), 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 \(x=4\)?

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 \(t=8\)?

\(\int\left(2e^{2x}+e\right)dx\)

\(\int{\sin{\left(9x\right)}+\cos{\left(9x\right)}dx}=\)

\(\int{x^3\left(x^4-2\right)^9}dx\)

Shown above is a slope field for which of the following differential equations?

Which of the following is the solution to the differential equation \(\frac{dy}{dx}=\frac{4x^3y}{x^4+1}\) whose graph contains the point \((0, 1)\)?

If \(0<c<1\), what is the area of the region enclosed by the graphs of \(y=0\), \(y=\frac{2}{x}\), \(x=1\), and \(x=c\)?

A particle moves along the x-axis with the velocity given by \(v\left(t\right)=24t^5+16t^3\) for time \(t\geq0\). If the particle is at position \(x=3\) 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 \(x=.5\), \(x=1.5\), \(x=3\), and \(x=4\). 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 \(a\left(t\right)=30\sqrt t\). The position of the particle is \(5\) when \(t=0\), and the position of the particle is \(10\) when \(t=1\). What is the velocity of the particle at \(t=0\)?