12/10 Related Rates

Front

What is a related rate problem in calculus?

Back

A related rate problem involves finding the rate at which one quantity changes with respect to another quantity that is also changing. These problems often involve the use of derivatives and the chain rule.

Front

How do you find the volume of a sphere?

Back

The volume V of a sphere with radius r is given by the formula V = (4/3)πr³.

Front

What is the formula for the area of a circle?

Back

The area A of a circle with radius r is given by the formula A = πr².

Front

What is the chain rule in calculus?

Back

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

How do you differentiate the volume of a sphere with respect to time?

Back

To differentiate the volume V = (4/3)πr³ with respect to time t, use the chain rule: dV/dt = dV/dr * dr/dt = 4πr² * (dr/dt).

Front

What is the relationship between the radius and the area of a circle?

Back

The area A of a circle increases with the square of the radius r. Specifically, A = πr², so if the radius increases, the area increases at a rate proportional to the square of the radius.

Front

How do you find the rate of change of the area of a circle when the radius is changing?

Back

To find the rate of change of the area A = πr² with respect to time, use the chain rule: dA/dt = 2πr * (dr/dt).

Front

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

Back

Units are crucial in related rates problems as they help ensure that the quantities being compared or calculated are consistent. Always check that the units match when applying formulas.

Front

How do you set up a related rates problem?

Back

To set up a related rates problem, identify the quantities involved, write down the relationships between them, differentiate with respect to time, and then solve for the desired rate.

Front

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

Back

The derivative of the volume V = (4/3)πr³ with respect to the radius r is dV/dr = 4πr².