Graphs of Functions & Their Derivatives (College Board AP® Calculus BC): Exam Questions

Let $g$ be a function with first derivative given by $g^{\text{'}}\left(x\right)={\int }_0^x{e}^{-2t}\mathit{d}t$. Which of the following must be true on the interval $0<x<3$?

The graph of the function $f$ is shown in the figure above. At which of the labeled points is $f^{\text{'}}\left(x\right)$ positive and increasing?

The volume of a cylindrical plastic container with a top and a bottom is to be $128\pi$ cubic centimeters. If a minimum amount of plastic is to be used to construct the container, what must be the height, in centimeters, of the container?

The graph of $f^{\text{'}}$, the derivative of the function $f$, is shown above.

Which of the following statements must be true?

I. $f$ has a relative minimum at $x=D$.

II. The graph of $f$ has a point of inflection at $x=C$.

III. The graph of $f$ is concave down for $A<x<0$.

Let $f$ be the function $f\left(x\right)=\sqrt{x}{e}^{kx}$, where $k$ is a constant. For what value of $k$ does $f$ have a critical point at $x=\frac{1}{4}$?

Let $f$ be a polynomial function with values of $f\text{'}\left(x\right)$ at selected values of $x$ given in the table above. Which of the following must be true for $-4<x<0$?

The graph of a twice-differentiable function $f$ is shown in the figure above. Which of the following is true?

The function $f$ is given by $f\left(x\right)=5-\frac{3}{4}{x}^4-3x^2$. On which of the following intervals is $f$ increasing?

The first derivative of the function $f$ is given by $f\text{'}\left(x\right)=1-x{\sin }^{2}x$. How many critical points does $f$ have on the open interval $\left(0,6\right)$?

The absolute minimum value of $f\left(x\right)=3x^3+18x^2-7$ on the closed interval $[-5,1]$ occurs at $x=$

Let $f$ be the function defined by $f\left(x\right)=\frac{\ln \left({\displaystyle \frac{1}{x}}\right)}{x^2},x>0$.

What is the absolute minimum value of $f$?

The graph of $f\text{'}$, the derivative of $f$, is the line shown in the figure above. If $f\left(0\right)=4$, then $f\left(-2\right)=$

Let $f$ be a twice-differentiable function with $f\text{'}\left(x\right)<0$ and $f\text{'}\text{'}\left(x\right)<0$ for all real numbers $x$, such that $f\left(2\right)=4$ and $f\left(4\right)=-4$. Of the following, which is a possible value for $f\left(6\right)$.

A particle moves along a straight line. The graph of the particle's position $x\left(t\right)$ at time $t$ is shown above for $0<t<18$. The graph has horizontal tangents at $t=4.5$ and $t=11$ and points of inflection at $t=8$ and $t=15$. For what values of $t$ is the velocity of the particle decreasing?

The function $f$ is continuous for $-5\le x\le 3$ and $f\left(-5\right)=f\left(3\right)=7$. If there is no $c$, where $-5<c<3$, for which $f\text{'}\left(c\right)=0$, which of the following statements must be true?

If $g$ is a differentiable function such that $g\left(x\right)>0$ for all real numbers $x$ and if $f\text{'}\left(x\right)=\left(9x^2-16\right)g\left(x\right)$, which of the following is true?

Let $f$ be a twice-differentiable function. Values of $f\text{'}$, the derivative of $f$, at selected values of $x$ are given in the table above. Which of the following statements must be true?

The function $f$ is continuous on the closed interval $\left[-2,1\right]$ and twice differentiable on the open interval $\left(-2,1\right)$. If $f\text{'}\left(0\right)=3$ and $f\text{'}\text{'}\left(x\right)>0$ on the open interval $\left(-2,1\right)$, which of the following could be a table of values for $f$?

Let $f$ be the function given by $f\left(x\right)=3x{e}^x$. The graph of $f$ is concave up when

The function $f$ has the property that $f\left(x\right)$ and $f\text{'}\text{'}\left(x\right)$ are positive for all real values $x$, and $f\text{'}\left(x\right)$ is positive only for $x>a$ where $a$ is a finite number. Which of the following could be the graph of $f$?