50 Question Ultimate AP Calculus AB Challenge (Calculator Required)

If \({lim}_{x\rightarrow a^-}{f(x)}={lim}_{x\rightarrow a^+}{f(x)}\) and \(f(a)=undefined\), then which of the following statements about \(f\) must be true?

Let \(f\) be a function that is continuous on the closed interval \([-2, 2]\) with \(f\left(-2\right)=7\) and \(f\left(2\right)=-7\). Which of the following statements must be true?

If \(f\) is a continuous function such that \(f\left(6\right)=2\), which of the following statements must be true?

The figure above shows the graph of a function \(f\) with domain \(-2\le x\le2\). Which of the following statements about \(f\) are true?
I. \({lim}_{x\rightarrow0^-}{f(x)}\) exists.
II. \({lim}_{x\rightarrow0^+}{f(x)}\) exists.
III. \({lim}_{x\rightarrow0}{f(x)}\)does not exist.

The graph of the function \(f\) is shown above. Which of the following limits does not exist?

The vertical line \(x=-4\) is an asymptote for the graph of the function \(f\). Which of the following statements must be false?

Let \(f\) be a differentiable function with selected values given in the table above. What is the average rate of change of \(f\) over the closed interval \(0\le x\le10\)?

The graph of the function \(h\) is shown in the figure above. Of the following, which has the greatest value?

The graphs of \(f\) and \(g\) are shown above. If \(h\left(x\right)=f\left(x\right)g(x)\), then \(h^\prime\left(1\right)=\)

The table above gives values of the differentiable functions \(f\) and \(g\) and their derivatives at \(x=1\). If \(h(x)=\frac{f(x)}{g(x)}\), what is the value of \(h^\prime(1)\)?

The radius of a sphere is decreasing at a rate of 3 centimeters per second. At the instant when the radius of the sphere is 5 centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface area \(S\) of a sphere with radius \(r\) is \(S=4\pi r^2\).)

The function \(f\) is continuous on the closed interval \([3, 5]\) and twice differentiable on the open interval \((3, 5)\). If \(f^\prime\left(4\right)=3\) and \(f^{\prime\prime}(x)>0\) on the open interval \((3, 5)\), which of the following could be a table of values for \(f\)?

In the \(xy\)-plane, the graph of the twice-differentiable function \(y=f(x)\) is concave down on the open interval \((1, 3)\) and is tangent to the line \(y=5x-3\) at \(x=2\). Which of the following statements must be true about the derivative of \(f\)?

Let \(y=f(x)\) define a twice-differentiable function and let \(y=r(x)\) be the line tangent to the graph of \(f\) at \(x=-4\). If \(r(x)\le f(x)\) for all real \(x\), which of the following must be true?

Let \(f\) be a twice-differentiable function on the open interval \((2, 7)\). If \(f^\prime\left(x\right)=0\) on \((a, b)\) and \(f^{\prime\prime}\left(x\right)=0\) on \((2, 7)\), which of the following could be the graph of \(f\)?

The function \(f\) is defined on the open interval \(0.1<x<1.7\) and has first derivative \(f^\prime\) given by \(f^\prime\left(x\right)=cos\left(x^2\right)\). Which of the following statements are true?
I. \(f\) has a relative maximum on the interval \(0.1<x<1.7\).
II. \(f\) has a relative minimum on the interval \(0.1<x<1.7\).
III. The graph of \(f\) has two points of inflection on the interval \(0.1<x<1.7\).

The first derivative of the function \(g\) is given by \(g^\prime\left(x\right)=-sin\left(\frac{\pi}{4}x^2\right)\) for \(-.5<x<1.7\). On which of the following intervals is \(g\) decreasing?

The number of people who have entered an amusement park is modeled by a function \(f(t)\), where \(t\) is measured in hours since the amusement park opened that day. The number of people who have left the amusement park since it opened that same day is modeled by a function \(g(t)\). If \(f^\prime\left(t\right)=150\left({1.04}^t\right)\) and \(g^\prime\left(t\right)=130+130cos\left(\frac{\pi(t-5)}{6}\right)\), at what time \(t\), for \(1\le t\le7\), is the number of people in the amusement park at a maximum?

The function \(P(t)\) models the population of the world, in billions of people, where \(t\) is the number of years since January 1, 2020. Which of the following is the best interpretation of the statement \(P^\prime(1)=1.277\)?

The first derivative of the function \(f\) is given by \(f^\prime(x)=cos\left(x^3\right)\). At which of the following values of \(x\) does \(f\) have a local maximum?

The function is continuous for \(-5\le x\le5\) and \(f\left(-5\right)=f\left(5\right)=3\). If there is no \(c\), where \(-5<x<5\), for which \(f^\prime\left(c\right)=0\), which of the following statements must be true?

A particle moves along a straight line with velocity given by \(v\left(t\right)=3-\left(1.03\right)^{-t^3}\) at time \(t\geq0\). What is the acceleration of the particle at time \(t=2\)?

The table gives selected values of the velocity, \(v(t)\), of a particle moving along the \(x\)-axis. At time \(t=0\), the particle is at the origin. Which of the following could be the graph of the position, \(x(t)\), of the particle for \(0\le t\le4\)?

A particle moves along a straight line with the velocity given by \(v\left(t\right)=5+e^\frac{t}{7}\) for time \(t\geq0\). What is the acceleration of the particle at time \(t = 3\)?

A particle moves along the \(x\)-axis so that its position at time \(t>0\) is given by \(x(t)\) and \(\frac{dx}{dt}=-35t^4+3t^2+2t\). The acceleration of the particle is zero when \(t=\)

The first derivative of the function \(f\) is defined by \(f^\prime(x)=sin\left(x-x^2\right)\) for \(-2\le x\le3\). On what intervals is \(f\) increasing?

The graph of \(f^\prime\), the derivative of \(f\), is shown above for \(0\le{x}\le5\). On what intervals is \(f\) decreasing?

The graph of the derivative of the function \(f\) is show in the figure above. The graph has horizontal tangent lines at \(x=-2\), \(x=-0.25\), and \(x=3\). At which of the following values of \(x\) does \(f\) have a relative maximum?

The graph of \(f^\prime\), the derivative of \(f\), is show above. The line tangent to the graph of \(f^\prime\) at \(x=0\) is vertical, and\(f^\prime\) is not differentiable at \(x=2\). Which of the following statements is true?

The derivative of the function \(f\) is given by \(f^\prime\left(x\right)=x^2cos\left(x^3\right)\). How many points of inflection does the graph of \(f\) have on the open interval \((-2,2)\)?

The figure above shows the graph of \(f’\), the derivative of a function \(f\), for \(0≤x≤3\). What is the value of \(x\) at which the absolute minimum of \(f\) occurs?

If \(\int_{4}^{-7}f\left(x\right)dx=-13\) and \(\int_{7}^{4}f\left(x\right)dx=-5\), what is the value of \(\int_{-7}^{7}{f(x)}dx\)?

If \(G(x)\) is an antiderivative for \(f(x)\) and \(G\left(1\right)=-2\), then \(G\left(5\right)=\)

The table above gives values of a function \(f\) and its derivative at selected values of \(x\). If \(f^\prime\) is continuous on the interval \([-5,-2]\), what is the value of \(\int_{-5}^{-3}{f^\prime\left(x\right)}dx\)?

The function \(f\) is continuous on the closed interval \([1,10]\). If \(\int_{1}^{10}{f(x)dx}=45\) and \(\int_{10}^{4}{f(x)dx}=-36\), then \(\int_{1}^{4}{3f(x)dx}=\)

If \(f^\prime(x)=cos\left(x^2\right)\) and \(f(3)=1\), then \(f(2)=\)

The graph of the function \(f\), which as a domain of \([0, 7]\), is shown in the figure above. The graph consists of a quarter circle of radius \(3\) and a segment with slope \(-1\). Let \(b\) be a positive number such that \(\int_{0}^{b}{f(x)}dx=0\). What is the value of \(b\)?

The derivative of the function \(f\) is continuous on the closed interval \([1, 5]\). Values of \(f\) and \(f^\prime(x)\) for selected values of \(x\) are given in the table above. If \(\int_{1}^{5}{f^\prime(t)}dt=13\), then \(f\left(5\right)=\)?

The graph of the piecewise linear function \(f\) is shown above. Let \(h\) be the function given by \(h(x)=\int_{-3}^{x}{f(t)}dt\). On which of the following intervals is \(h\) increasing?

The height above the ground of a person riding a roller coaster \(t\) seconds after the ride begins is modeled by the differentiable function \(R\), where \(R(t)\) is measured in feet. Which of the following is an interpretation of the statement \(R^\prime\left(5.5\right)=-13.597\)?

A slope field for a differential equation is shown in the figure above. If \(y=f(x)\) is particular solution to the differential equation through the point \((0, 0)\) and \(h\left(x\right)=5x\bullet f(x)\), then \(h^\prime\left(0\right)=\)

A garden is located beside a home has rectangular boundary as shown in the figure above. The worm density of the soil at any point along a strip \(x\) feet from the home’s edge is \(f(x)\) worms per square foot. Which of the following expressions gives the worm population of the garden?

An electric tractor uses electricity at the rate \(g\left(t\right)=3+2cos\left(\frac{t}{100}\right)\) kilowatts per minute, where \(t\) is the number of minutes since starting the tractor. To the nearest kilowatt, what is the total amount of kilowatts used from \(t = 0\) to \(t = 45\) minutes?

The rate at which honey is leaking from a beehive is modeled by the function \(B\) defined by \(B(t)=2+sin\left(t^3\right)\) for time \(t\geq0\). \(B(t)\) is measured in liters per hour, and \(t\) is measured in hours. How much honey leaks out of the beehive during the first 90 minutes?

What is the area enclosed by the curves \(y={x}^3-{8x}^2+13x\) and \(y=x+3\)?

What is the area of the region enclosed by the graphs of \(y=\sqrt{16x-x^4}\) and \(y=\frac{x}{2}\)?

What is the average value of \(y=\frac{\cos(x)}{x^2+2x-2}\) on the closed interval \([-5, 3]\)?

The temperature \(F\), in degrees Fahrenheit (\(°F\)), of a cup of coffee \(t\) minutes after it is poured is given by \(F(t)=68+115e^{-0.083t}\). To the nearest degree, what is the average temperature of the coffee between \(t=0\) and \(t=7\) minutes?

An object is traveling in a straight line has position \(x(t)\) at time \(t\). If the initial position is \(x\left(0\right)=5\) and the velocity of the object is \(v\left(t\right)=\sqrt[4]{3+t^2}\), what is the position of the object at time \(t=5\)?

A particle moves along a straight line for 8 seconds so that its velocity, in meters per second, is modeled by the graph shown above. During the time interval \(0\le t\le8\), what is the total distance the particle travels?

Let \(R\) be the region bounded by the graphs of \(y=e^x\), \(y=e^2\), and \(x=0\). Which of the following gives the volume formed by revolving \(R\) about the line \(y=1\)?