Rates of Change & Related Rates (College Board AP® Calculus AB): Exam Questions

A particle moves along the $x$-axis with its position at time $t$ given by $x\left(t\right)=\left(2t-a\right)\left(t-b\right)$, where $a$ and $b$ are non-zero constants and $\frac{a}{2}\ne b$.

For which of the following values of $t$ is the particle at rest?

For $t\ge 0$, the position of a particle moving along the $x$-axis is given by $x\left(t\right)=\cos t-\sin t$. What is the velocity of the particle at the point where the acceleration is first equal to 0?

The radius of a sphere is changing at a rate directly proportional to the radius, where the constant of proportionality is $k$.

At the instance when the radius of the sphere is 4 centimeters, what is the rate of change, in terms of $k$, of the surface area of the sphere? (The surface area $S$ of a sphere with radius $r$ is $S=4\pi r^2$.)

The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is $25\pi$ meters?

Let $y=\sin x\times \ln x$. Both $x$ and $y$ vary with time in such a way that $x$ increases at the constant rate of $2\pi$ units per second. The rate at which $y$ is changing when $x=\frac{\pi }{2}$ is

A security light on the wall of a bank shines on a fleeing criminal . The light is at a height of 4 meters above the ground, the criminal is 1.75 meters tall and is running at a speed of 6 meters per second. What is the rate at which the criminal's shadow is lengthening?

The maximum acceleration attained on the interval $0\le t\le 4$ by the particle whose velocity is given by $v\left(t\right)=-\frac{1}{3}{t}^3+t^2+8t+5$ occurs when $t=$

The graph below shows the displacement, $x$, of a particle in meters in the interval $0\le t\le 4$ seconds.

Which of the following tables could be the table of values for the velocity, $v$, in meters per second for the particle in the interval $1\le t\le 3$?

If the base $b$ of a triangle is increasing at a rate of 12 centimeters per second while its height $h$ is decreasing at a rate 12 centimeters per second, which of the following must be true about the area $A$ of the triangle?

The radius of a right circular cylinder is increasing at a rate of 4 units per second. The height of the cylinder is decreasing at a rate of 6 units per second.

Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius $r$ and height $h$ of the cylinder?

(The volume of a cylinder with radius $r$ and height $h$ is $V=\pi r^{2}h$.)

A particle moves along the $x$-axis so that at any time $t\ge 0$, its velocity is given by $v\left(t\right)=4t+4\cos \left(2t\right)$. At what time, where $t>0$, is the acceleration equal to zero for the second time?

The diagonal $c$ of the rectangle shown is increasing at the rate of 3 centimeters per second. The rate of change of length $a$ is twice the rate of change of length $b$. At what rate is the length $b$ increasing when $a=5$ cm and $b=12$ cm ?

Let $y=3e^{\cos \left(4x\right)}$. Both $x$ and $y$ vary with time in such a way that $y$ increases at the constant rate of 6 units per second. The rate at which $x$ is changing when $x=\frac{\pi }{8}$ is