Related Rates Flashcard

Related rates involve finding the rate at which one quantity changes in relation to another quantity that is also changing. It often involves using derivatives to relate the rates of change of different variables.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): $$c^2 = a^2 + b^2$$ .

The volume V of a sphere with radius r is given by the formula: $$V = \frac{4}{3}\pi r^3$$ .

To find the rate of change of volume with respect to radius, differentiate the volume formula: $$\frac{dV}{dr} = 4\pi r^2$$ .

The area A of a circle with radius r is given by the formula: $$A = \pi r^2$$ .

To find the rate of change of area with respect to radius, differentiate the area formula: $$\frac{dA}{dr} = 2\pi r$$ .

The volume V of a cone is given by the formula: $$V = \frac{1}{3} \pi r^2 h$$ .

To find the rate of change of height with respect to volume, use implicit differentiation on the volume formula.

Negative rates indicate that a quantity is decreasing over time, such as the height of water in a tank or the volume of air in a balloon.

The chain rule is used to relate the rates of change of different variables. For example, if V is a function of r, and r is a function of t, then $$\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}$$ .