Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Relativistic quantum mechanics wikipedia , lookup
Human height wikipedia , lookup
Renormalization group wikipedia , lookup
Time value of money wikipedia , lookup
Simplex algorithm wikipedia , lookup
Okishio's theorem wikipedia , lookup
Interest rate wikipedia , lookup
Related Rates and Applications Lesson 3.7 1 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 2 Expanding Ripple • At the point in time when r=8 – radius is increasing at 3 in/sec – That is we are given r=8 dr 3 dt • We seek the rate that the area is changing at that specific time – We want to know dA dt View Spreadsheet demonstration 3 Solution Strategy 1. Draw a figure label with variables do NOT assign exact values unless they never change in the problem A r 2. Find formulas that relate the variables A r 2 dr 3 dt 4 Solution Strategy 3. Differentiate the equation with respect to time dA dr 2 r dt dt 4. Substitute in the given information r 8 dr 3 dt 2 8 3 48 in / sec 2 5 Example • Given x y 25 2 2 dx 4 dt dy • Find when x = 3 dt Note: we must differentiate implicitly with respect to t dx dy 2x 2 y 0 dt dt 6 Example • Now substitute in the things we know – dx 4 dt x=3 • Find other values we need – when x = 3, 32 + y2 = 25 y=4 and dx dy 2x 2 y 0 dt dt 7 Example dx dy 2x 2 y 0 dt dt • Result dy 244 24 0 dt dy 32 4 dt 8 8 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 9 R1 Electricity • The combined electrical R resistance R of R1 and R2 1 1 1 connected in parallel is R R1 R2 given by • R1 and R2 are increasing at rates of 1 and 1.5 ohms per second respectively. • At what rate is R changing when R1 = 50 and R2 = 75? 2 10 Draining Water Tank • Radius = 20, Height = 40 • 1 2 Volume r h 3 • The flow rate = 80 gallons/min • What is the rate of change of the radius when the height = 12? dV 80 dt dr ?? dt 11 Draining Water Tank • At this point in time the height is fixed 1 2 Volume r 12 3 • Differentiate implicitly with respect to t, dV 1 dr 2 r 12 • Substitute in known dt 3 dt values • Solve for dr/dt 12 Assignment • Lesson 3.7 • Page 187 • Exercises 1 – 7 odd, 13 – 27 odd 13