Functions & Rates of Change (College Board AP® Precalculus): Exam Questions

The graph of a function $f$ is shown in the $xy$-plane.

What are the zeros of $f$?

The table below gives selected values of a function $f$.

The values of $f$ are increasing as $x$ increases. Which of the following statements about the graph of $f$ for $0\le x\le 4$ is true?

The figure shows the graph of a function $g$. The zero and extrema for $g$ are labeled, and the point of inflection of the graph of $g$ is labeled. Let $A$, $B$, $C$, $D$, and $E$ represent the $x$-coordinates at those points.

Of the following, on which interval is $g$ decreasing and the graph of $g$ concave up?

A water tank is being filled using a hose. The rate at which the volume of water in the tank increases is constant when the hose is running. During the first three hours, the hose runs at a reduced flow rate due to low water pressure. The hose is then turned off while a valve is adjusted. After the adjustment, the hose runs at its full flow rate until the tank is completely full. The entire process of filling the tank takes eight hours. Which of the following graphs could depict this situation, where time, $t$ in hours, is the independent variable, and the volume of water in the tank, $V$ in gallons, is the dependent variable?

The depth of water, in metres, at a tidal monitoring station is modelled by a function $W$. The graph of $y=W\left(t\right)$ is shown for $0\le t\le 40$, where $t$ is the number of hours since midnight. What are all intervals of $t$ on which the depth of water is increasing at a decreasing rate?

The average temperature, in degrees Celsius, in a city on day $t$ of the year is modelled by the function $T$ defined by

$T\left(t\right)=12\cos \left(\frac{2\pi }{365}(t-200)\right)+10$

Which of the following correctly describes the average temperature on day $t=320$?

A delivery van leaves a depot to make deliveries. For the first 2 hours, the van travels slowly through city streets. The van then stops for 1 hour to unload packages at a delivery site. After completing the delivery, the van travels on the motorway for 4 hours. The van ends 100 km from the depot, and the entire journey takes 7 hours.

Which of the following graphs could represent this situation, where $t$ is the time in hours and $D$ is the distance from the depot in kilometres?

The graph of a function $g$ is shown in the $xy$-plane. Four points $A$, $B$, $C$, and $D$, with coordinates $A\left(0,1\right)$, $B\left(2,5\right)$, $C\left(4,7\right)$, and $D\left(6,8\right)$, are labeled on the graph.

On which of the following intervals is the average rate of change of $g$ the greatest?

The table gives values for a function $w$ at selected values of $x$.

Which of the following statements about the concavity of the graph of $w$ on the interval $0\le x\le 8$ is true based on the data?

The table gives values for a function $f$ at selected values of $x$ near $x=3$.

Based on the table, which of the following is the best approximation for the rate of change of $f$ at $x=3$?

The graph of a function $f$, with domain $\left[-5,5\right]$, is shown in the $xy$-plane.

On which of the following intervals is the function $f$ increasing?

The function $p$ models the population of a town, in thousands, $t$ years after 2010. The table gives values for $p\left(t\right)$ at selected values of $t$ near $t=5$ and $t=10$.

Based on the table, which of the following statements comparing the rate of change of $p$ at $t=5$ to the rate of change of $p$ at $t=10$ is true?

A relation $k$ between two sets of real numbers is described as follows: for every input value $x$, the output value $y$ is the absolute value of $x$.

Which of the following correctly identifies whether $k$ is a function and gives a valid set of ordered pairs $\left(x,y\right)$ that is consistent with the description of the relation?

The table below gives selected values of a function $f$.

Assume that between any two consecutive values of $x$ listed in the table, $f$ is either always increasing or always decreasing, and the rate of change of $f$ is either always increasing or always decreasing.

On which of the following intervals is $f$ positive and increasing, and the graph of $f$ concave up?

The graph of the function $f$ is concave down on the closed interval $\left[0,4\right]$. The values $f\left(0\right)=2$, $f\left(2\right)=8$, and $f\left(4\right)=k$ are given. Which of the following is a possible value of $k$?

For the function $f\left(x\right)=\frac{1}{x}$, the average rate of change of $f$ on the interval $\left[2,c\right]$ is $-\frac{1}{10}$. What is the value of $c$?