AP DC Calculus AB - Related Rates Final Exam Review

Front

What is a related rates problem in calculus?

Back

A related rates problem involves finding the rate at which one quantity changes with respect to another quantity that is also changing, often using derivatives.

Front

Define the term 'implicit differentiation'.

Back

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, allowing us to find dy/dx without solving for y explicitly.

Front

What is the formula for the volume of a sphere?

Back

V = (4/3)πr³, where r is the radius of the sphere.

Front

How do you find the rate of change of the volume of a sphere with respect to its radius?

Back

Use the formula dV/dr = 4πr², which gives the rate of change of volume with respect to radius.

Front

What is the relationship between the rates of change of the height and volume of a cone?

Back

For a cone, dV/dt = (1/3)πr²(dh/dt) + (2/3)πhr(dh/dt), where r is the radius and h is the height.

Front

Explain the concept of 'related rates' using a ladder problem as an example.

Back

In a ladder problem, if a ladder slides down a wall, the rate at which the foot of the ladder moves away from the wall can be found using the Pythagorean theorem and related rates.

Front

What is the formula for the volume of a cone?

Back

V = (1/3)πr²h, where r is the radius and h is the height.

Front

How do you determine the rate at which the height of water in a tank is changing?

Back

Use the formula dh/dt = (dV/dt) / (πr²), where dV/dt is the rate of volume change and r is the radius of the tank.

Front

What is the significance of the negative sign in related rates problems?

Back

A negative sign indicates that the quantity is decreasing, such as the height of water in a tank.

Front

How do you apply the chain rule in related rates problems?

Back

The chain rule is applied to relate the rates of change of different variables, allowing us to express one rate in terms of another.