50 Question Ultimate AP Calculus AB Challenge (No Calculator)

If \(f\left(x\right)=x^{5}+5x\), then \(\frac{d}{dx}\left(f\left(\cos{x}\right)\right)=\)

\( f\left(x\right)=\left\{\begin{array}{ll} -3ax+b,\quad if\ {x}<{-3} \\ x^2-ax, \quad if\ x\geq-3\end{array}\right.\)

Let \(f\) be the function defined above, where \(a\) and \(b\) are constants. If \(f\) is differentiable at \(x=-3\), what is the value of \(a+b\)?

The graph of the function \(f\) shown above has horizontal tangent lines at \(x=.5\), \(x=1.5\), \(x=2.5\), and \(x=3.5\). 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?

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 \(\left(f^{-1}\right)^\prime\left(11\right)\)?

If \(f\left(x\right)=x^{12}+12x\), then \(\frac{d}{dx}\left(f\left(\ln{x}\right)\right)=\)

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

Avian flu spreads among a population of \(P\) chickens at a rate inversely proportional to the product of the number of chickens who have the disease and the number of chickens who do not have the disease. If \(c\) denotes the number of chickens 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?

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?

\(\lim_{x\rightarrow-\infty}{\frac{(3x-1)(2+x)}{(x-2)(x+4)}=}\)

A particle moves along a straight line. The graph of the particles position \(x(t)\) at time \(t\) is show above for \(0<t<4\). The graph has horizontal tangents at \(t=1\) and \(t=3\) and a point of inflection at \(t=2\). For what values of t is the velocity of the particle decreasing?

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

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 \(\left(f^{-1}\right)^\prime\left(7\right)\)?

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

\( f\left(x\right)=\left\{\begin{array}{ll} -ax+b,\quad if\ {x}<{0} \\ x^2-ax, \quad if\ x\geq0\end{array}\right.\)

Let \(f\) be the function defined above, where \(a\) and \(b\) are constants. If \(f\) is differentiable at \(x=0\), what is the value of \(a+b\)?

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?

\(f(x)=\sqrt x+\frac{3}{\sqrt x}\), then \(f^\prime(1)=\)

\(\int_{1}^{0}\frac{{8x}^3}{\sqrt[3]{7x^4+1}}dx\)

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

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

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

\( f\left(x\right)=\left\{\begin{array}{ll} ax+b,\quad if\ {x}<{0} \\ x^2-ax, \quad if\ x\geq0\end{array}\right.\)

Let \(f\) be the function defined above, where \(a\) and \(b\) are constants. If \(f\) is differentiable at \(x=0\), what is the value of \(b-a\)?

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

Let \(f\) be the function defined by \(f\left(x\right)=x^3+2x^2+x\). Let \(g\left(x\right)=f^{-1}(x)\), where \(g\left(4\right)=1\). What is the value of \(g^\prime(4)\)?

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

\( f\left(x\right)=\left\{\begin{array}{ll} x^2cos\left(\frac{\pi}{2}x\right),\quad for\ x<2 \\ x^2+cx-8, \quad for\ x\geq2\end{array}\right.\)

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

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

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

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

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

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

\(\int{\sin{\left(7x\right)}+\cos{\left(7x\right)}dx}=\)

The points \((2, 0)\), \((x, 0)\), \(\left(x,\frac{1}{x^2}\right)\), and \(\left(2,\frac{1}{x^2}\right)\) are the vertices of a rectangle, where \(x\geq2\), as shown in the figure above. For what value of \(x\) does the rectangle have a maximum area?

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

\( f\left(x\right)=\left\{\begin{array}{ll} ax+b,\quad if\ {x}<{0} \\ x^2-ax, \quad if\ x\geq0\end{array}\right.\)

Let \(f\) be the function defined above, where \(a\) and \(b\) are constants. If \(f\) is differentiable at \(x=0\), what is the value of \(a-b\)?

If \(f\left(x\right)=x^{2}+2x\), then \(\frac{d}{dx}\left(f\left(\sin{x}\right)\right)=\)

The function \(f\) is defined by \(f(x)={x}^3-{x}^2+1\). The application of the Mean Value Theorem to \(f\) on the interval \(-1\le x\le1\) guarantees the existence of a value of \(c\), where \(-1<c<1\), such that \(f^\prime(c)=\)

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

\(\int{{5x}^2\left(x^3-8\right)^4}dx\)

A particle moves along a straight line. The graph of the particles position \(x(t)\) at time \(t\) is show above for \(0<t<4\). The graph has horizontal tangents at \(t=1\) and \(t=3\) and a point of inflection at \(t=2\). For what values of t is the velocity of the particle decreasing?

The graph of the function \(f\) shown above has horizontal tangent lines at \(x=.5\), \(x=1.5\), \(x=2.5\), and \(x=3.5\). 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?

If \(f\left(x\right)=e^{\left(\frac{7}{x^2}\right)}\), then \(f^\prime\left(x\right)=\)

\(\int\frac{x^5}{x^6-7}dx\)

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

If \(\frac{dy}{dt}=-5e^{-\frac{t}{2}}\) and \(y(0)=10\), what is the value of \(y(4)\)?

The base of a solid is the region bounded by the \(x\)-axis and the graph of \(y=\sqrt{9-{9x}^2}\). For the solid, each cross-section perpendicular to the \(x\)-axis is a square. What is the volume of the solid?

Which of the following is the solution to the differential equation \(\frac{dy}{dx}=\frac{x^4}{y}\) with the initial condition \(y\left(0\right)=2\)?

If \(e^{xy}-y^2=e-9\), then at \(x=\frac{1}{3}\) and \(y=3\), \(\frac{dy}{dx}=\)

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 increasing on the interval \((-.5,.5)\)?

If \(y=sinx\ cosx\), then at \(x=\frac{\pi}{2}\), \(\frac{dy}{dx}=\)

\(\lim_{x\rightarrow\infty}{\frac{\sqrt{64x^6+9}}{25x^3+5}}\) is

A disease spreads among a population of \(P\) wolves at a rate 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?