#4 Sample Exam AP Calculus AB Multiple Choice (Timed-No Calculator Allowed)

\(\lim_{x\rightarrow\infty}{\frac{\left(3x-1\right)\left(2-7x^2\right)}{\left(x-2\right)\left(8x^2+4\right)}=}\)

If \(f\left(x\right)=x^{9}+9x\), then \(\frac{d}{dx}\left(f\left(\cos{x}\right)\right)=\)

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

If \(f\left(x\right)=e^{\left(\frac{5}{x^2}\right)}\), then \(f^\prime\left(x\right)=\)

If \(y=\left(\frac{2x}{7x+1}\right)^4\), then \(\frac{dy}{dx}=\)

\( f\left(x\right)=\left\{\begin{array}{ll} \frac{x^2-4}{x-2},\quad if\ x\neq2 \\ 4, \quad if\ x=2\end{array}\right.\)

Let \(f\) be the function defined above. Which of the following statements about \(f \) are true?

I. \(f \) has a limit at \(x=2\).

II. \(f \) is continuous at \(x=2\).

III. \(f \) is differentiable at \(x=2\).

Let \(f\) be the differentiable function such that \(f\left(2\right)=13,\ f\left(5\right)=2,\ f^\prime\left(2\right)=-4,\ and\ f^\prime\left(5\right)=3\). The function \(g\) is differentiable and \(g\left(x\right)=f^{-1}(x)\) for all \(x\). What is the value of \(g^\prime(2)\)?

\(\frac{d}{dx}\left({sin}^{-1}x+2\sqrt x\right)=\)

If \(\sin{\left(xy\right)}=x^8\), then \(\frac{dy}{dx}=\)

If \(\sin{\left(7xy\right)}=x\), then \(\frac{dy}{dx}=\)

What is the slope of the line tangent to the curve \(y=\arctan(3x)\) at the point at which \(x=-\frac{1}{3}\)?

If \(f\left(x\right)=\cos\left(4x\right)\), then \(f^\prime\left(\frac{\pi}{24}\right)\)

The first derivative of the function \(f\) is given by \(f^\prime\left(x\right)=-5x^4+20x^3\). What are the \(x\)-coordinates of the points of inflection of the graph of \(f\)?

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?

Let \(f\) be a function with a second derivative given by \(f^{\prime\prime}\left(x\right)={(x-2)}^2{(x-4)}^3{(x-5)}^4\). What are the \(x\)-coordinates of the points of inflection of the graph of \(f\)?

The table above gives selected values for a twice-differentiable function \(f\). Which of the following must be true?

Let \(f\) be the function given by \(f\left(x\right)=x^3+4x\). For what value of \(x\) in the closed interval \([2, 5]\) does the instantaneous rate of change of \(f\) equal the average rate of change of \(f\) on that interval?

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 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 graph of the function \(f\) is shown above for \(0\le\ x\le4\). Of the following, which has the greatest value?

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 \(t=10\)?

\(\int{\frac{7}{x^4}dx}=\)

\(\int{\sin{\left(13x\right)}+\cos{\left(13x\right)}dx}=\)

\(\int_{2}^{0}\frac{3x^2}{\sqrt{x^3+1}}dx\)

A disease spreads among a population of \(P\) wolves at a rate inversely proportional to the product of the number of wolves who have the disease and the number of wolves who do not have the disease. If \(w\) denotes the number of wolves 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 \(\frac{dy}{dx}=\frac{8x^7y}{x^8+1}\) whose graph contains the point \((0, 1)\)?

Let \(f\) be the function defined by \(f\left(x\right)=\frac{1}{x}\). What is the average value of \(f\) on the interval \([5, 7]\)?

A particle moves along the x-axis with the velocity given by \(v\left(t\right)=12t^5+24t^3\) for time \(t\geq0\). If the particle is at position \(x=1\) at time \(t=0\), what is the position of the particle at time \(t=1\)?

If \(y=f(x)\) is a solution to the differential equation \(\frac{dy}{dx}=e^{x^4}\) with the initial condition \(f\left(0\right)=3\), which of the following is true?

A particle moves along the \(x\)-axis so that at time \(t\geq0\), the acceleration of the particle is \(a\left(t\right)=91\sqrt[6]{t}\). The position of the particle is \(3\) when \(t=0\), and the position of the particle is \(45\) when \(t=1\). What is the velocity of the particle at \(t=0\)?