Use implicit differentiation W-up Use implicit differentiation

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 𝑑𝑦 𝑑𝑡 𝑤ℎ𝑒𝑛 𝑥=2, 𝑦=−3 𝑎𝑛𝑑 𝑑𝑥 𝑑𝑡 =3 Taking the derivative with respect to “t” – there are no “t’s” in the equation When you take the derivative multiply by 𝑑 (𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆) 𝑑𝑡 (just like implicit differentiation) Product rule for xy Substitute in given values Simplify and solve for 𝑑𝑦 𝑑𝑡 Factored out a 𝑑𝑦 𝑑𝑡

Steps for solving related rate problem Draw a picture (if possible) Identify / assign the variables Identify what you want_____ when____ List what is known, rates Write formula that relates variables in problem Differentiate Substitute numerical values for the variables and rate 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 𝒅𝑨 𝒅𝒕 when r = 30 feet 3) rate of change of radius 𝒅𝒓 𝒅𝒕 = .5 ft/sec 4) A = pr2 5) Differentiate 𝒅𝑨 𝒅𝒕 =𝟐𝝅𝒓 𝒅𝒓 𝒅𝒕 6) 𝒅𝑨 𝒅𝒕 =𝟐𝝅 𝟑𝟎 (.𝟓) 7) Solve 𝒅𝑨 𝒅𝒕 =𝟑𝟎𝝅≈𝟗𝟒.𝟓 𝒇𝒕𝟐/𝒔𝒆𝒄𝒐𝒏𝒅 Remember: When you take the derivative multiply by 𝑑 (𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆) 𝑑𝑡 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 𝒅𝑨 𝒅𝒕 when r = 3 meters 3) rate change of volume or 𝒅𝑽 𝒅𝒕 = 10m3/ min 4) A = 4pr2 and V = 𝟒 𝟑 pr3 5) Differentiate 𝒅𝑨 𝒅𝒕 =𝟖𝝅𝒓 𝒅𝒓 𝒅𝒕 𝒅𝑽 𝒅𝒕 =𝟒𝝅𝒓𝟐 𝒅𝒓 𝒅𝒕 6) 𝒅𝑨 𝒅𝒕 =𝟖𝝅 𝟑 𝒅𝒓 𝒅𝒕 oh no we don’t know 𝒅𝒓 𝒅𝒕 , what can we use to find it?

𝑑𝑟 𝑑𝑡 = 10 36𝜋 = 5 18𝜋 now plug, into formula in step 6 𝟏𝟎=𝟒𝝅𝒓𝟐 𝒅𝒓 𝒅𝒕 𝟏𝟎=𝟒𝝅(𝟑)𝟐 𝒅𝒓 𝒅𝒕 I want the rate when radius is 3, solve for 𝒅𝒓 𝒅𝒕 𝑑𝑟 𝑑𝑡 = 10 36𝜋 = 5 18𝜋 now plug, into formula in step 6 𝒅𝑨 𝒅𝒕 =𝟖𝝅 𝟑 5 18𝜋 7) Solve 𝒅𝑨 𝒅𝒕 = 𝟐𝟎 𝟑 ≈𝟔.𝟔𝟕𝒎𝟐/𝒎𝒊𝒏𝒖𝒕𝒆

Homework: 14.6 # 1-8 all, 9,11,13