Implicit Differentiation and Related Rates: Solving Problems with Rates of Change
Explore implicit differentiation and related rates through examples of finding rates of change in variables such as volume, area, and surface area. Learn step-by-step methods for solving related rate problems involving variables and their rates in real-world scenarios like balloons, ripples in ponds, and spheres. Understand the concepts through practical examples and visual aids.
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W-UP USE IMPLICIT DIFFERENTIATION + = + = + 2 2 4 4 2 1. 9 2. 5 2 1 x y x xy y x 3 4 1 4 xy 5 4 x x y y = = 1. ' 2. ' y y 3 20 y
14.6 RELATED RATES SWBAT SOLVE RELATED RATE PROBLEMS Problems involving rates of related variables are related rate problems. Example: The rate at which the volume of a balloon is changing at a specific radius
Ex1: If xy+ 6x + y3= -2 find ?? ??? ?? ? = 2,? = 3??? ?? ??= 3 Taking the derivative with respect to t there are no t s in the equation When you take the derivative multiply by ? (????????) (just like implicit differentiation) ?? dy dt dx dt dx dt dy dt Product rule for xy + + + = 2 6 3 0 x y y Substitute in given values dy dt dy dt Simplify and solve for ?? 3 3 6 3 3( 3) + + + = 2 2 0 ?? Factored out a ?? dy dt dy dt dy dt ?? 9 27 + + = 2 0 9 dy dt = (2 27) + = 9 29
STEPS FOR SOLVING RELATED RATE PROBLEM 1) Draw a picture (if possible) Identify / assign the variables 2) Identify what you want_____ when____ 3) List what is known, rates 4) Write formula that relates variables in problem 5) Differentiate 6) Substitute numerical values for the variables and rate 7) Solve.
A child throws a stone into a still pond causing a circular ripple to spread. If the radius of the circle increases at the constant rate of 0.5 feet/ second, how fast is the area of the ripple increasing when the radius is 30 feet? 1) r = radius, A = area , t = seconds 2) Want rate area is increasing or ?? when r = 30 feet 3) rate of change of radius ?? 4) A = r2 ?? ?? = .5 ft/sec Remember: When you take the derivative multiply by ? (????????) 5) Differentiate ?? ??= ????? ?? ?? 6) ?? ??= ?? ?? (.?) 7) Solve?? Must label answer ??= ??? ??.? ???/??????
A balloon in the form of a sphere is being inflated at the rate of 10 cubic meters per minute. Find the rate at which the surface area of the sphere is increasing at the instant when the radius of the sphere is 3 meters. 1) r = radius, A = area , V = volume, t = minutes 2) Want rate area is increasing or ?? 3) rate change of volume or ?? 4) A = 4 r2 and V = ? ?? when r = 3 meters ?? = 10m3/ min ? r3 5) Differentiate ?? ??= ????? ?? ??= ?????? ?? ?? 6) ?? ?? ??oh no we don t know ?? ??= ?? ? ??, what can we use to find it?
?? = ?????? ?? ?? = ??(?)??? ??I want the rate when radius is 3, solve for ?? ?? ?? ?? = 10 5 18?now plug, into formula in step 6 36?= ?? ??= ?? ? 5 18? 7) Solve ?? ??=?? ? ?.????/??????
