Section 2.6: Related Rates

Introduction to Related Rates We have seen a lot of relations (whether implicit or explicit) that involve two variables (frequently x and y). It is possible these two variables are themselves functions of another variable, such as t. For instance:

Introduction to Related Rates Let’s investigate what occurs when t changes: t x y Equation As x and y change, their rates of change are related to each other. But how are they related? Notice how when t changes, both the x and y change in relation to the value of t.

Introduction to Related Rates In order to take the derivative of the relation using x and y , it must be done with the respect to t. For instance: Differentiate both sides Chain Rule Twice Now we know how the rate of change for x and y are related to each other. In our exercises, we will not need to know the exact relations.

Example 1 Suppose x and y are both differentiable functions of t and are related by . Find when x = 10, if when x = 10. Find the derivative by differentiating both sides. Chain Rule Substitute the known information Solve for the unknown

Example 2 Suppose x and y are both differentiable functions of t and are related by . Find when x = 9, if when x = 9 and y>0. Find the derivative by differentiating both sides. Find other important values: x Chain Rule Substitute the known information Solve for the unknown

Example 3 A spherical balloon is being filled with a gas in such a way that when the radius is 2ft, the radius is increasing at the rate 1/6 ft/min. How fast is the volume ( ) changing at this time? Find the derivative by differentiating both sides. Chain Rule Substitute the known information ft3 per minute Solve for the unknown

Related Rates Guidelines Draw a figure, if appropriate, and assign variables to the quantities that vary. (Be careful not to label a quantity with a number unless it never changes in the problem) Find a formula or equation that relates the variables. (Eliminate unnecessary variables) Differentiate the equations. (typically implicitly) Substitute specific numerical values and solve algebraically for any required rate. (The only unknown value should be the one that needs to be solved for.)

Example 1 A person 6 ft tall is walking away from a streetlight 20 ft high at the rate of 7 ft/s. At what rate is the length of the person’s shadow increasing? Find the rates by differentiating both sides. 20 ft 6 ft Chain Rule y x Substitute the known information Using similar triangles, the equation is: Solve for the unknown ft/s

Example 2 A bag is tied to the top of a 5 m ladder resting against a vertical wall. Suppose the ladder begins sliding down the wall in such a way that the foot of the ladder is moving away from the wall. How fast is the bag descending at the instant the foot of the ladder is 4 m from the wall and the foot is moving away at the rate of 2 m/s? Find the rates by differentiating both sides. 5 m y Ladder Chain Rule Substitute the known information x Using The Pythagorean Theorem, the equation is: Find other important values: Solve for the unknown m/s x

Example 3 A trough 10 ft long has a cross section that is an isosceles triangle 3 ft deep and 8 ft across. If water flows in at the rate 2 ft3/min, how fast is the surface rising when the water is 2 ft deep? Find the rates by differentiating both sides. 10 ft 8 ft Nothing is known about b… b 3 ft h Using similar triangles: Chain Rule Substitute the known information Using the volume of a prism, the equation is: Solve for the unknown ft/min

Example 4 A rocket launches with a velocity of 550 miles per hour. 25 miles away there is a photographer filming the launch. At what rate is the angle of elevation of the camera changing when the rocket achieves an altitude of 25 miles? Find the rates by differentiating both sides. This is “x” and there is no “x” in the derivative… x Chain Rule Substitute the known information Θ 25 mi Using The Trigonometry, the equation is: Use “x” to find other important values: Solve for the unknown rad/h