50 MCQ Ultimate AP Calc AB Challenge (Calculator Required)

The function \(f\) is defined on the open interval \(2.6<x<3.4\) 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 \(2.6<x<3.4\).
II. \(f\) has a relative minimum on the interval \(2.6<x<3.4\).
III. The graph of \(f\) has two points of inflection on the interval \(2.6<x<3.4\).

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

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 \(t=4\) minutes?

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?

The graph of the derivative of the function \(f\) is show in the figure above. The graph has horizontal tangent lines at \(x=-1\), \(x=1\), and \(x=3\). At which of the following values of \(x\) does \(f\) have a relative maximum?

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 \((0, 2)\) and \(h\left(x\right)=5x\bullet f(x)\), then \(h^\prime\left(0\right)=\)

Let \(f\) be a twice-differentiable function on the open interval \((3, 7)\). If \(f^\prime\left(x\right)<0\) on \((a, b)\) and \(f^{\prime\prime}\left(x\right)=0\) on \((3, 7)\), which of the following could be the graph of \(f\)?

The vertical line \(x=7\) is an asymptote for the graph of the function \(f\). Which of the following statements must be false?

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 \(h\) decreasing?

Let \(R\) be the region bounded by the graphs of \(y=e^x\), \(y=e^4\), and \(x=0\). Which of the following gives the volume formed by revolving \(R\) about the line \(y=-2\)?

A particle moves along a straight line with velocity given by \(v\left(t\right)=5-\left(1.02\right)^{-t^3}\) at time \(t\geq0\). What is the acceleration of the particle at time \(t=2\)?

The first derivative of the function \(g\) is given by \(g^\prime\left(x\right)=sin\left(\pi x^2\right)\) for \(-.5<x<1.5\). On which of the following intervals is \(g\) decreasing?

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 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 \(t=8\) minutes?

The function \(f\) is continuous on the closed interval \([1,11]\). If \(\int_{1}^{11}{f(x)dx}=33\) and \(\int_{11}^{3}{f(x)dx}=-12\), then \(\int_{1}^{3}{2f(x)dx}=\)

What is the average value of \(y=\frac{\cos(x)}{x^2-3x-5}\) on the closed interval \([-1, 1]\)?

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

In the \(xy\)-plane, the graph of the twice-differentiable function \(y=f(x)\) is concave up on the open interval \((2, 4)\) and is tangent to the line \(y=-7x+3\) at \(x=3\). Which of the following statements must be true about the derivative of \(f\)?

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 \(f\)?

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 \(f\)?

In the \(xy\)-plane, the graph of the twice-differentiable function \(y=f(x)\) is concave up 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 \(f\)?

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 \(-4\le t\le0\)?

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

The radius of a sphere is decreasing at a rate of 2 centimeters per second. At the instant when the radius of the sphere is 3 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 \(S=4\pi r^2\).)

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 \(t = 10\) minutes?

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

The function \(f\) is continuous on the closed interval \([1,9]\). If \(\int_{1}^{9}{f(x)dx}=13\) and \(\int_{9}^{5}{f(x)dx}=36\), then \(\int_{1}^{5}{2f(x)dx}=\)

The function \(f\) is continuous on the closed interval \([1,9]\). If \(\int_{1}^{9}{f(x)dx}=13\) and \(\int_{9}^{5}{f(x)dx}=-36\), then \(\int_{1}^{5}{2f(x)dx}=\)

What is the average value of \(y=\frac{\cos(x)}{x^2+2x-2}\) on the closed interval \([1, 5]\)?

If \(\int_{-3}^{2}f\left(x\right)dx=-7\) and \(\int_{6}^{2}f\left(x\right)dx=-2\), what is the value of \(\int_{-3}^{6}{f(x)}dx\)?

What is the area enclosed by the curves \(y={x}^4-{8x}^3+{13x}^2+2\) and \(y=x-10\)?

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?

A home uses water at the rate \(w\left(t\right)=10+7sin\left(\frac{t}{30}\right)\) liters per day, where \(t\) is the number of days from the beginning of the year. To the nearest liter, what is the total amount of water used from \(t = 0\) to \(t = 50\) days?

What is the area enclosed by the curves \(y={x}^5-{x}^3+{x}^2+2\) and \(y=5x+3\)?

Let \(f\) be a function that is continuous on the closed interval \([0, 2]\) with \(f\left(0\right)=7\) and \(f\left(2\right)=16\). Which of the following statements must be true?

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

Let \(f\) be a twice-differentiable function on the open interval \((3, 7)\). If \(f^\prime\left(x\right)<0\) on \((a, b)\) and \(f^{\prime\prime}\left(x\right)<0\) on \((3, 7)\), which of the following could be the graph of \(f\)?

Let \(f\) be a twice-differentiable function on the open interval \((3, 7)\). If \(f^\prime\left(x\right)>0\) on \((a, b)\) and \(f^{\prime\prime}\left(x\right)=0\) on \((3, 7)\), which of the following could be the graph of \(f\)?

If \({lim}_{x\rightarrow a^-}{f(x)}=3,{lim}_{x\rightarrow a^+}{f(x)}=3,\ and\ f(a)=3\), then which of the following statements about \(f\) must be true?

If \(f^\prime(x)=cos\left(x^2\right)\) and \(f(3)=4\), then \(f(2)=\)

The first derivative of the function \(g\) is given by \(g^\prime\left(x\right)=sin\left(\frac{\pi}{2}x^2\right)\) for \(-.5<x<1.5\). On which of the following intervals is \(g\) decreasing?

The graphs of \(f\) and \(g\) are shown above. If \(h\left(x\right)=f\left(x\right)g(x)\), then \(h^\prime\left(4\right)=\)

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

The vertical line \(x=7\) is an asymptote for the graph of the function \(f\). Which of the following statements must be false?

A wilderness area located beside a canyon has rectangular boundary as shown in the figure above. The bear density of the wilderness area at any point along a strip \(x\) miles from the canyon’s edge is \(f(x)\) bear per square mile. Which of the following expressions gives the bear population of the wilderness area?

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 \(t=3\) minutes?

The graph of the derivative of the function \(f\) is show in the figure above. The graph has horizontal tangent lines at \(x=-1\), \(x=1\), and \(x=3\). At which of the following values of \(x\) does \(f\) have a relative minimum?

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

If \(f^\prime(x)=cos\left(x^2\right)\) and \(f(3)=5\), then \(f(2)=\)

A park located beside a stream has rectangular boundary as shown in the figure above. The mosquito density of the park at any point along a strip \(x\) miles from the stream’s edge is \(f(x)\) mosquitos per square mile. Which of the following expressions gives the mosquito population of the park?