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

\({lim}_{x\rightarrow3}{\frac{x^2-9}{x^2-2x-15}}\)

Let \(f\) be the function given by \(f(x)=\frac{(2x-5)(4x+7)}{{(x-5)}^2}\). If the line \(y=b\) is a horizontal asymptote to the graph of \(f\), then \(b=\)?

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

What is the average rate of change of \(y=sin(2x)\) on the interval \(\left[0,\frac{5\pi}{6}\right]\)?

If \(f\left(x\right)=e^{\left(\frac{9}{x}\right)}\), then \(f^\prime\left(x\right)=\)

If \(f\left(x\right)=x^4+4x\), then \(\frac{d}{dx}\left(f\left(\ln{x}\right)\right)=\)

\( f\left(x\right)=\left\{\begin{array}{ll} \frac{x^2-36}{x-6},\quad if\ x\neq6 \\ 12, \quad if\ x=6\end{array}\right.\)

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 \( x=6\).

\( f\left(x\right)=\left\{\begin{array}{ll} 6x-4,\quad for\ x<1 \\ ln(6x-5), \quad for\ x\geq1\end{array}\right.\)
Let \(f\) be the function defined above. Which of the following statements about \(f\) are true?
I. \({lim}_{x\rightarrow1^-}{f(x)}={lim}_{x\rightarrow1^+}{f(x)}\)
II. \({lim}_{x\rightarrow1^-}{f^\prime(x)}={lim}_{x\rightarrow1^+}{f^\prime(x)}\)
III. \(f\) is differentiable at \(x=1\).

The table above gives selected values for a differentiable and decreasing function \(f\) and its derivative. If \(f^{-1}\) is the inverse function of \(f\), what is the value of \(\left(f^{-1}\right)^\prime\left(13\right)\)?

\({lim}_{h\rightarrow0}{\frac{sin\left(\frac{2\pi}{3}+h\right)-sin\left(\frac{2\pi}{3}\right)}{h}}\)

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

The top of a 15-foot-long ladder rests against a vertical wall with the bottom of the ladder on level ground, as shown above. The ladder is sliding down the wall at a constant rate of 24 feet per second. At what rate, in radians per second, is the acute angle between the bottom of the ladder and the ground changing at the instant the bottom of the ladder is 9 feet from the base of the wall?

Let \(f\) be the function given by \(f(x)=\frac{kx}{4x^2+1}\), where \(k\) is a constant. For what values of \(k\), if any, is \(f\) strictly decreasing on the interval \((-.5,.5)\)?

If \(f^{\prime\prime}(x)=x\left(x+3\right)^2\), then the graph of \(f\) is concave down for

Let \(f\) be the function with derivative defined by \(f^\prime(x)=x^3-9x\). At which of the following values of \(x\) does the graph of \(f\) have a point of inflection?

The function \(f\) is defined by \(f(x)={2x}^3-{4x}^2+1\). The application of the Mean Value Theorem to \(f\) on the interval \(1\le x\le2\) guarantees the existence of a value of \(c\), where \(1<c<2\), such that \(f^\prime(c)=\)

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 \(t=.25\), \(t=1\), and \(t=2\) and a point of inflection at \(t=.5\), \(t=1.5\), and \(t=2.5\). For what values of t is the velocity of the particle decreasing?

The graph of \(y=f(x)\) on the closed interval \([0,6]\) is shown above. Which of the following could be the graph of \(y=f'(x)\)?

The slope of the tangent line to the graph of \(y=\ln(2-3x)\) at \(x=-2\) 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 \(f\left(3.9\right)\) using the line tangent to the graph of \(f\) at \(x=4\)?

The values of a continuous function \(f\) for selected values of \(x\) are given in the table above. What is the value of the right Riemann sum approximation \(\int_{0}^{20}{f(x)dx}\) using the subintervals \([0, 3]\), \([3, 12]\), and \([12, 20]\)?

What are all values of \(x\) for which \(\int_{x}^{3}t^2dt\) is equal to 0?

\( \int{3e^{2x}+\frac{2}{x}dx} \)

If \(\frac{dy}{dt}=-20e^{-\frac{t}{2}}\) and \(y(0)=40\), what is the value of \(y(2)\)?

\( \int{x^2\left(x^3+7\right)^8}dx= \)

Which of the following is the solution to the differential equation \(\frac{dy}{dx}=5y\) with the initial condition \(y(0)=4\)?

An object moves along a straight line so that at any time \(t\geq0\) its velocity is given by \(v(t)=4sin(3t)\). What is the distance traveled by the object from \(t=0\) to the first time that it stops?

Which of the following is an equation for the line tangent to the graph of \(y=3+\int_{1}^{x}{e^{{-t}^4}dt}\) at the point where \(x=1\)?

The graph of the function \(f\) shown above has horizontal tangent lines at \(x=.5\), \(x=1.5\), \(x=2.5\), and \(x=3.5\). 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 base of a solid is the region bounded by the \(x\)-axis and the graph of \(y=\sqrt{4-x^2}\). For the solid, each cross-section perpendicular to the \(x\)-axis is a square. What is the volume of the solid?