AP Calculus AB Chapter 2, Section 5

AP Calculus ABChapter 2, Section 6Related Rates2013 - 2014

Related rates𝑑𝑦𝑑𝑥 We have used the chain rule to find implicitly. Anotherimportant use of the chain rule is to find the rates of changeof two or more related variables that are changing withrespect to time.

Related Rates For example: If water is draining out of a conical tank, whatare the variables that are changing with respect to time?

Conical Tank continued Use implicit differentiation for the volume of a cone anddescribe what each piece means.

Two Rates That Are Related Suppose x and y are both differentiable functions of t and𝑑𝑦2are related by the equation 𝑦 𝑥 3. Find when𝑥 1, given that𝑑𝑥𝑑𝑡𝑑𝑡 2 when 𝑥 1.

Ripples in a Pond A pebble is dropped into a calm pond, causing ripplesin the form of concentric circles. The radius r of theouter ripple is increasing at a constant rate of 1 footper second. When the radius is 4 feet, at what rate isthe total area A of the disturbed water changing?

Guidelines for Solving Related-Rate Problems1. Identify all given quantities and quantities to bedetermined. Make a sketch and label the quantities.2. Write an equation involving the variables.3. Use the Chain Rule, implicitly differentiate both sides ofthe equation with respect to time t.4. After completing step 3, substitute into the resultingequation all known values for the variables and their ratesof change. Solve if necessary.

Examples of mathematical modelsinvolving rates of changeVerbal StatementThe velocity of a car after traveling for 1hour is 50 miles per hourWater is being pumped into a swimmingpool at a rate of 10 cubic meters per hourA gear is revolving at a rate of 25revolutions per minute (1 revolution 2πrad)Mathematical Model

An Inflating Balloon Air is being pumped into a spherical balloon at a rate of 4.5cubic feet per minute. Find the rate of change of the radiuswhen the radius is 2 feet.

The Speed of an Airplane Tracked by Radar An airplane is flying on a flight path that will take it directlyover a radar tracking station. If s is decreasing at a rate of 400miles per hour when s 10 miles, what is the speed of theplane?

A Changing Angle of Elevation A television camera at ground level is filming the lift-off of aspace shuttle that is rising vertically according to theposition equation 𝑠 50𝑡 2 , where s is measured in feetand t is measured in seconds. The camera is 2000 ft fromthe launch pad. Find the rate of change in the angle ofelevation of the camera shown at 10 seconds after lift-off.

The Velocity of a Piston In the engine shown on page 153, a 7-inch connecting rodis fastened to a crank of radius 3 inches. The crankshaftrotates counterclockwise at a constant rate of 200revolutions per minute. Find the velocity of the piston𝜋when 𝜃 .3

2.6 Homework Pg. 154 – 156, #’s: 3, 7, 15, 19, 27, 31, 43 7 problems

Conical Tank continued . Examples of mathematical models involving rates of change Verbal Statement Mathematical Model The velocity of a car after traveling for 1 hour is 50 miles per hour Water is being pumped i