related rates

Front

What are related rates in calculus?

Back

Related rates are a method used in calculus to find the rate at which one quantity changes in relation to another quantity that is also changing.

Front

Define the term 'instantaneous rate of change'.

Back

The instantaneous rate of change of a function at a point is the limit of the average rate of change as the interval approaches zero, often represented as the derivative.

Front

What is the formula for the volume of a sphere?

Back

V = (4/3)πr³, where V is the volume and r is the radius.

Front

How do you find the derivative of a function with respect to time?

Back

Use implicit differentiation, treating all variables as functions of time.

Front

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

Back

As the radius increases, the volume increases at a rate proportional to the cube of the radius.

Front

If air is leaving a balloon at a constant rate, how does that affect the radius?

Back

The radius decreases as the volume decreases, and the rate of change of the radius can be found using related rates.

Front

What is the formula for the area of a circle?

Back

A = πr², where A is the area and r is the radius.

Front

How do you calculate the rate of change of the area of a circle as the radius changes?

Back

Use the formula dA/dt = 2πr(dr/dt), where dr/dt is the rate of change of the radius.

Front

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

Back

Negative rates indicate a decrease in the quantity being measured, such as a shrinking radius or volume.

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.