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

\( f\left(x\right)=\left\{\begin{array}{ll} x^2cos\left(\frac{\pi}{2}x\right),\quad for\ x<2 \\ x^2+cx-10, \quad for\ x\geq2\end{array}\right.\)

Let \(f\) be the function defined above, where \(c\) is a constant. For what value of \(c\), if any, is \(f\) continuous at \(x=2\)?

\({lim}_{x\rightarrow-4}{\frac{x^2-16}{x^2-2x-8}}\)

What are all horizontal asymptotes of the graph of \(y=\frac{3+5^x}{2-5^x}\) in the \(xy\)-plane?

Let \(f\) be the function given by \(f(x)=\frac{(3x-5)(4x+7)}{{(2x-5)}^2}\). If the line \(y=b\) is a horizontal asymptote to the graph of \(f\), then \(b=\)?

What is the average rate of change of \(y=cos(2x)\) on the interval \(\left[0,\frac{\pi}{4}\right]\)?

If \(f(x)=x^{10}+7\) and \(g\) is a differentiable function of \(x\), what is the derivative of \(f(g(x))\)?

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

\( f\left(x\right)=\left\{\begin{array}{ll} 7x-5,\quad for\ x<1 \\ ln(7x-6), \quad for\ x\geq1\end{array}\right.\)
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 \(\left(f^{-1}\right)^\prime\left(4\right)\)?

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

If \(y=sinx\ cosx\), then at \(x=0\), \(\frac{dy}{dx}=\)

If \(y^9+y^8=x^2\), then \(\frac{dy}{dx}=\)

A person is laying on the ground 6 meters away from where a hot air balloon is being launched as shown above. The hot air balloon is rising at a rate of 50 meters per second. At what rate, in radians per second, is the acute angle between the persons line of site and the ground changing at the instant the bottom of the hot air balloon is 8 meters from the ground?

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

\(f(x)=\sqrt x+\frac{3}{\sqrt x}\), then \(f^\prime(9)=\)

The velocity, \(v\), in meters per second, of a certain type of wave is given by \(v(h)=3\sqrt h\), where \(h\) is the depth, in meters, of the water through which the wave moves. What is the rate of change, in meters per second per meter, of the velocity of the wave with respect to the depth of the water, when the depth is 5 meters?

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 \(f^\prime(c)=\)

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 values of a continuous function \(f\) for selected values of \(x\) are given in the table above. What is the value of the left Riemann sum approximation \(\int_{0}^{60}{f(x)dx}\) using the subintervals \([0, 20]\), \([20, 40]\), and \([40, 60]\)?

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

If \(\frac{dy}{dt}=-5e^{-\frac{t}{2}}\) and \(y(0)=10\), what is the value of \(y(8)\)?

Which of the following is an antiderivative of \(8\ {sec}^2x+10\)?

\( \int{x^3\left(x^4+7\right)^8}dx= \)

Which of the following is the solution to the differential equation \(\frac{dy}{dx}=5y^2\) with the initial condition \(y(0)=7\)?

If \(0<c<1\), what is the area of the region enclosed by the graphs of \(y=0\), \(y=\frac{3}{x}\), \(x=1\), and \(x=c\)?

An object moves along a straight line so that at any time \(t\geq0\) its velocity is given by \(v(t)=2sin(3t)\). 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 \(y=3+\int_{-1}^{x}{e^{{t}^3}dt}\) at the point where \(x=-1\)?

The base of a solid is the region bounded by the \(x\)-axis and the graph of \(y=-\sqrt{16-x^2}\). For the solid, each cross-section perpendicular to the \(x\)-axis is a square. What is the volume of the solid?