AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.6: Related Rates

What you’ll learn about  Related Rate Equations  Solution Strategy  Simulating Related Motion …and why Related rate problems are at the heart of Newtonian mechanics; it was essentially to solve such problems that calculus was invented.

What are Related Rate Equations? Any equation involving two or more variables that are differentiable functions of time “t” can be used to find an equation that relates their corresponding rates. We use implicit differentiation to differentiate several variables with respect to time.

Strategy for Solving Related Rate Problems  To solve related rates problems: 1. Draw a picture. 2. Identify a variable whose rate of change you seek.

Strategy for Solving Related Rate Problems  Solving related rates problems (cont) 3.Find a formula relating the variable whose rate of change you seek with one or more variables whose rate of change you know. This is the hard part. The formulas can come from geometry, physical laws, or wherever. Sometimes the AP test will supply the formula, but don’t count on it. It is important to keep variables variable. Never plug in a number which can “freeze the picture” until after differentiating.

Strategy for Solving Related Rate Problems  Solving related rates problems (cont) 4.Differentiate implicitly with respect to time t. Remember all those rules (product, quotient, chain, etc.). It is tempting to forget them when dealing with variables like r and h, which do not look like functions of time. 5.Plug and chug. It is safe now to “freeze the picture.” 6.Make sure that you have answered the problem. Write out your answer in a sentence (with units) to see that it makes sense.

Example: Finding Related Rate Equations a) Assume that the radius r of a sphere is a differentiable function of t and let V be the volume of the sphere. Find an equation that relates dV/dt and dr/dt. V = dV/dt = b) Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt. V = dV/dt = You try: Given x 2 + y 2 = z 2. Find an equation that relates dx/dt, dy/dt, and dz/dt.

Example 1 – Related Rates  Ex 1: 1. Draw a picture. 2. Identify a variable whose rate of change you seek. dy/dt= -2 ft/sec x y dx/dt=? 25 ft

Example 1 (cont’d) – Related Rates 3. Find a formula relating the variable whose rate of change you seek with one or more variables whose rate of change you know. 4. Differentiate implicitly with respect to time t.

Example 1 (cont’d) – Related Rates 5. Plug and chug (Freeze the picture)

Example 2 – Related Rates  Ex 2: 1. Draw a picture. 2. Identify a variable whose rate of change you seek. Leaking out at a rate of 5 cubic cm/sec So dV/dt=-5 9 cm h 3 cm r At what rate was the height of water changing when h=4 cm ?

Example 2 (cont’d) – Related Rates 3. Find a formula relating the variable whose rate of change you seek with one or more variables whose rate of change you know.

Example 2 (cont’d) – Related Rates 4.Differentiate implicitly with respect to time t.

Example 2 (cont’d) – Related Rates 5.Plug and chug (Freeze the picture)

Example 2: A hot-air balloon rising straight up from a level field is tracked by a range finder 500’ from the lift off point. At the moment the range finder’s elevation angle is π/4, the angle is increasing at the rate of 0.14 radians per minutes. How fast is the balloon rising at that moment? 1. Identify variables 2. Draw picture – label 3. Find formula 4. Differentiate implicitly 5. Substitute explicit values into differentiated formula 6. Interpret solution in a sentence

Example 3 - A Highway Chase

Example 3 A police cruiser, approaching a right-angles intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car? 1. Identify variables 2. Draw picture – label 3. Find formula 4. Differentiate implicitly 5. Substitute explicit values into differentiated formula 6. Interpret solution in a sentence

Example 3 - A Highway Chase (cont’d)

You try Exercise 13 An Airplane is flying at an altitude of 7 mi and passes directly over a radar antenna as shown in the figure on p 251. When the plane is 10 mi from the antenna (s = 10), the radar detects that the distance s is changing at the rate of 300 mph. What is the speed of the airplane at that moment? 1 & 2 Identify variables / Draw picture – label 3. Find formula 4. Differentiate implicitly 5. Substitute explicit values into differentiated formula 6. Interpret solution in a sentence

Example 4 Water runs into a conical tank at the rate of 9 ft 3 /min. The tank stands point down and has a height of 10’ and a base radius of 5’. How fast is the water level rising when the water is 6’ deep? 1. Identify variables 2. Draw picture – label 3. Find formula 4. Differentiate implicitly 5. Substitute explicit values into differentiated formula 6. Interpret solution in a sentence (note 2 nd solution strategy on p 249)

What are related rate equations and what can they tell us? You Tube Just Math Tutoring Related Rate Equations Examples 1 & 2

In Review: Strategy for Solving Related Rate Problems

In Review: Strategy for Solving Related Rate Problem