1 Related Rates and Applications Lesson 3.7

2 General vs. Specific Note the contrast … General situation –properties true at every instant of time Specific situation –properties true only at a particular instant of time We will consider a rock dropped into a pond … generating an expanding ripple

3 Expanding Ripple At the point in time when r = 8 –radius is increasing at 3 in/sec –That is we are given We seek the rate that the area is changing at that specific time –We want to know r = 8 View Spreadsheet demonstration View Spreadsheet demonstration

4 Solution Strategy 1.Draw a figure label with variables do NOT assign exact values unless they never change in the problem 2.Find formulas that relate the variables A r

5 Solution Strategy 3.Differentiate the equation with respect to time 4.Substitute in the given information

6 Example Given Find when x = 3 Note: we must differentiate implicitly with respect to t

7 Example Now substitute in the things we know – x = 3 Find other values we need –when x = 3, y 2 = 25 and y = 4

8 Example Result

9 Guidelines for Related-Rate Problems 1.Identify given quantities, quantities to be determined Make a sketch, label quantities 2.Write equation involving variables 3.Using Chain Rule, implicitly differentiate both sides of equation with respect to t 4.After step 3, substitute known values, solve for required rate of change

10 Electricity The combined electrical resistance R of R 1 and R 2 connected in parallel is given by R 1 and R 2 are increasing at rates of 1 and 1.5 ohms per second respectively. At what rate is R changing when R 1 = 50 and R 2 = 75? R1R1 R1R1 R2R2 R2R2

11 Draining Water Tank Radius = 20, Height = 40 The flow rate = 80 gallons/min What is the rate of change of the radius when the height = 12?

12 Draining Water Tank At this point in time the height is fixed Differentiate implicitly with respect to t, Substitute in known values Solve for dr/dt

13 Assignment Lesson 3.7 Page 187 Exercises 1 – 7 odd, 13 – 27 odd