AP Calculus AB Semester Exam Review

Front

What is the formula for the volume of a cube?

Back

The volume of a cube is given by the formula V = s^3, where s is the length of an edge.

Front

If the volume of a cube is changing, how does it affect the edge length?

Back

The rate of change of the volume is related to the rate of change of the edge length by the formula: \( \frac{dV}{dt} = 3s^2 \frac{ds}{dt} \).

Front

What is the relationship between the volume of a sphere and its radius?

Back

The volume of a sphere is given by the formula V = \( \frac{4}{3} \pi r^3 \), where r is the radius.

Front

How do you find the rate of change of the radius of a sphere when its volume is changing?

Back

Use the formula: \( \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} \).

Front

What is the formula for the area of a rectangle?

Back

The area A of a rectangle is given by A = l * w, where l is the length and w is the width.

Front

How do you minimize the amount of fencing needed for two equal rectangular lots?

Back

To minimize fencing, set up the area constraint and use calculus to find the dimensions that minimize the perimeter.

Front

What is the derivative of the volume of a cube with respect to time?

Back

The derivative is \( \frac{dV}{dt} = 3s^2 \frac{ds}{dt} \).

Front

What is the derivative of the volume of a sphere with respect to time?

Back

The derivative is \( \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} \).

Front

What is the significance of related rates in calculus?

Back

Related rates problems involve finding the rate at which one quantity changes in relation to another, often using derivatives.

Front

How do you set up a related rates problem?

Back

Identify the quantities involved, write down the relationships between them, and differentiate with respect to time.