Related Rates + Other Review

The area A of a circle is given by the formula A = $$\frac{1}{2} \times \text{radius}^2$$ .

In related rates problems, the rate of change of the area A with respect to time t is related to the rate of change of the radius r with respect to time t by the formula $$\frac{dA}{dt} = 2\text{π}r \frac{dr}{dt}$$ .

The volume V of a sphere is given by the formula V = $$\frac{4}{3}\text{π}r^3$$ . The rate of change of volume with respect to time is $$\frac{dV}{dt} = 4\text{π}r^2 \frac{dr}{dt}$$ .

A function has a horizontal tangent line at a point if the derivative at that point is zero, i.e., $$f'(x) = 0$$ .

If f'(x) changes from negative to positive at x, then f(x) has a local minimum at that point.

The Pythagorean theorem states that in a right triangle, $$a^2 + b^2 = c^2$$ , where c is the hypotenuse. In related rates, it helps relate the rates of change of the sides of the triangle.

Use the Pythagorean theorem to relate the height of the ladder on the wall and the distance from the wall, then differentiate with respect to time.

The volume V of a sphere is given by the formula V = $$\frac{4}{3}\text{π}r^3$$ .

To find the derivative, use implicit differentiation: $$2y \frac{dy}{dx} - (6x^2 + 24x) = 0$$ .

The rate of change of the area A with respect to time t is given by $$\frac{dA}{dt} = 2\text{π}r \frac{dr}{dt}$$ .