1. Use implicit differentiation
W-up
Use implicit differentiation

2. 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

3. 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

4. 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.

5. 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

6. 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?

7. 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 = 𝜋 = 5 18𝜋 now plug, into formula in step 6
𝒅𝑨 𝒅𝒕 =𝟖𝝅 𝟑 5 18𝜋
7) Solve 𝒅𝑨 𝒅𝒕 = 𝟐𝟎 𝟑 ≈𝟔.𝟔𝟕𝒎𝟐/𝒎𝒊𝒏𝒖𝒕𝒆

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