50 Question Ultimate AP Calculus AB Challenge (No Calculator)

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

Let \(f\) be the function given by \(f(x)=\frac{(8x-3)(x-4)}{{(x-8)}^2}\). If the line \(y=b\) is a horizontal asymptote to the graph of \(f\), then \(b=\)?

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

\( f\left(x\right)=\left\{\begin{array}{ll} x^2cos\left(\frac{\pi}{2}x\right),\quad for\ x<2 \\ x^2+cx-6, \quad for\ x\geq2\end{array}\right.\)

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

\({lim}_{x\rightarrow-5}{\frac{x^2-25}{x^2-3x-10}}\)

\(\lim_{x\rightarrow-\infty}{\frac{\sqrt{25x^6+7}}{9x^3+6}}\) is

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

If \(f(x)=(x+8){(x^2+2)}^6\), then \(f^\prime\left(x\right)=\)

If \(f\left(x\right)=e^{\left(\frac{1}{x}\right)}\), then \(f^\prime\left(x\right)=\)

\(\frac{d}{dx}\left({cos}^{-1}x+4\sqrt x\right)=\)

\(\frac{d}{dx}\left({sin}^{-1}x+4\sqrt x\right)=\)

If \(f\left(x\right)=x^{10}+10x\), then \(\frac{d}{dx}\left(f\left(\ln{x}\right)\right)=\)

If \(f(x)=(x-8){(x^2+2)}^6\), then \(f^\prime\left(x\right)=\)

If \(y=2\cos{\left(7x\right)}\), then \(\frac{dy}{dx}=\)

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

If \(y^8+y^7=x^2\), then \(\frac{dy}{dx}=\)

If \(\sin{\left(9xy\right)}=x\), then \(\frac{dy}{dx}=\)

If \(\sin{\left(11xy\right)}=x\), 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 \(6\pi\) meters, what is the rate of change of the volume of the pile, in cubic meters per hour?

If \(e^{xy}-y^2=e-16\), then at \(x=-\frac{1}{4}\) and \(y=-4\), \(\frac{dy}{dx}=\)

Let \(f\) be the function with derivative defined by \(f^\prime(x)=x^3-4x\). At which of the following values of \(x\) does the graph of \(f\) have a point of inflection?

Let \(f\) be the function with derivative given by \(f^\prime\left(x\right)=\frac{-5x^2}{\left(7+x^2\right)^2}\). On what interval is \(f\) decreasing?

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 \((-.5,.5)\)?

The slope of the tangent line to the graph of \(y=\ln(2-3x)\) at \(x=-3\) is

\(f(x)=\sqrt x+\frac{4}{\sqrt x}\), then \(f^\prime(4)=\)

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 \(f\)?

In the \(xy\)-plane, the line \(-14x+y=k\), where \(k\) is a constant, is tangent to the graph of \(y=3x^2+2x+2\). What is the value of \(k\)?

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 4 meters?

What is the slope of the line tangent to the curve \(y=\arcsin(2x)\) at the point at which \(x=\frac{1}{4}\)?

The velocity, \(v\), in meters per second, of a certain type of wave is given by \(v(h)=4\sqrt h\), 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 7 meters?

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 \(f\) decreasing?

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

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

\(f(x)=\sqrt x+\frac{4}{\sqrt x}\), then \(f^\prime(0)=\)

Let \(f\) be the function with derivative defined by \(f^\prime(x)=x^3-18x\). At which of the following values of \(x\) does the graph of \(f\) have a point of inflection?

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 \(f^{\prime\prime}\left(x\right)={(x-2)}^3{(x-4)}^2(x-5)\). 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?

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 \(0\le x\le3\) guarantees the existence of a value of \(c\), where \(0<c<3\), such that \(f^\prime(c)=\)

The points \((-3, 0)\), \((x, 0)\), \(\left(x,\frac{1}{x^2}\right)\), and \(\left(-3,\frac{1}{x^2}\right)\) are the vertices of a rectangle, where \(x\leq{-3}\), as shown in the figure above. For what value of \(x\) does the rectangle have a maximum area?

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 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 least value?

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 \(\int_{0}^{30}{f(x)dx}\) using the subintervals \([0, 10]\), \([10, 25]\), and \([25, 30]\)?

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 \([12, 20]\)?

\(\int{\sin{\left(11x\right)}+\cos{\left(11x\right)}dx}=\)

Which of the following is an antiderivative of \(9\ {sec}^2x+11\)?

\( \int{6e^{2x}+\frac{3}{x}dx} \)

\(\int\frac{x^2}{{3x}^3-6}dx\)

\(\int\frac{{4x}^3}{x^4-5}dx\)

If \(\frac{dy}{dt}=-20e^{-\frac{t}{2}}\) and \(y(0)=40\), what is the value of \(y(6)\)?

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

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

Which of the following is the solution to the differential equation \(\frac{dy}{dx}=x^3y\) with the initial condition \(y\left(0\right)=-2\)?

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 \((0, 1)\)?

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

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 noon. Which of the following gives the meaning of \(\int_{1}^{7}{f(t)}dt\) in the context described?

A particle moves along the \(x\)-axis so that at time \(t\geq0\), the acceleration of the particle is \(a\left(t\right)=66\sqrt[5]{t}\). The position of the particle is \(5\) when \(t=0\), and the position of the particle is \(39\) when \(t=1\). What is the velocity of the particle at \(t=0\)?

A particle moves along the x-axis with the velocity given by \(v\left(t\right)=8t^3+2t\) for time \(t\geq0\). If the particle is at position \(x=2\) at time \(t=0\), what is the position of the particle at time \(t=1\)?

A particle moves along the \(x\)-axis so that at time \(t\geq0\), the acceleration of the particle is \(a\left(t\right)=90\sqrt[4]{t}\). The position of the particle is \(5\) when \(t=0\), and the position of the particle is \(39\) when \(t=1\). What is the velocity of the particle at \(t=0\)?

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?

The graph of the function \(f\) shown above has horizontal tangent lines at \(x=1\) and \(x=4.5\). 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?

Let \(h\) be the function defined by \(h\left(x\right)=\int_{2\pi}^{x}{{cos}^2t}dt\). Which of the following is an equation for the line tangent to the graph of \(h\) at the point where \(x=2\pi\)?