Day 1 intro to related rates notes including group activity.notebook March 10, 2017

One application of implicit differentiation is "Related Rates." Related rates tell us how

different elements of an equation are changing with respect to each other.

2.6

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

Intro to related rates

Actual Example

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

In groups of three (when possible):You will be assigned one of the following scenarios. You are to (1) illustrate the situation, (2) identify which quantities are changing and which are not, and (3) choose an appropriate geometric formula which applies to the situation.

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

A. A biplane is flying at an altitude of 6000 meters. It is 10,000 meters from your home (which is on the ground of course). You are interested in knowing the plane's speed so that you can calculate how quickly the distance between the plane and your home is changing.

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

B. It is the fifth day that school has been cancelled due to snow and ice. And the huge spherical snowball that you made three days earlier is slowly melting. Assume it is melting in such a way that it retains its spherical shape :) You are interested in finding out how quickly it is melting so you can determine the rate of change of its radius.

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

C. Your dad has told you that you must paint the shutters of all the second story windows on your home (thanks Dad!). You lean a 30 foot ladder against your home such that the base of the ladder is 8 feet from the base of your home. But then tragedy strikes, the ladder begins to slide down the wall! You are interested in knowing the rate at which the base of the ladder is moving away from the base of the wall so you can calculate the rate at which the top of the ladder is sliding down the wall, so you will know how fast you will be traveling when you hit the ground.

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

D. You are lucky enough to have a beautiful cylindrical above ground pool in your backyard. It has a depth of 5 feet and a diameter of 20 feet. Since it is almost spring time you are filling the pool with water from the hose. You are interested in knowing the rate at which the height of the water is rising so you can calculate the rate at which the water is flowing into the pool.

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

E. At your new job, you have a water cooler with those stupid conical paper cups to drink out of. Well someone thought it was time to haze the new guy and so they put a small hole in the bottom of your cup. Once you fill it up, it begins to slowly leak. You are interested in knowing the rate at which the height of the water left in your cup is decreasing so that you can calculate the rate at which the water is leaking from your cup.

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

RELATED RATES

Cars leaving an intersection example:

What's an equation that fits this situation?

Label the variables, from the equation, in our illustration.

Which variables in the equation are changing as functions of time and which ones are not changing (if any).

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

Find

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

Check out this website

http://people.hofstra.edu/stefan_waner/RealWorld/tutorials/frames4_4.html

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

Solving a Related Rates Problem(Steps from website)

Step 1: Identify the changing quantities, possibly with the aid of a sketch. 

Step 2: Write down an equation that relates the changing quantities. 

Step 3: Differentiate both sides of the equation with respect to t. 

Step 4: Go through the whole problem and restate it in terms of the quantities and their rates of change. Rephrase all statements regarding changing quantities using the phrase "the rate of change of . . . ." 

Last Step: Substitute the given values in the derived equation you obtained above, and solve for the required quantity. 

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

An acorn is tossed into a pond, causing ripples to form. The radius of the outer-most ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the ripple changing?

What do we know? What's constant? What changes?

What do we want to know?

1.  Illustrate.  Make sure to label!

2.  Write down your known/unknown quantities.  Make sure to note the ones that are changing.

3.  Write an equation that relates each of the variables.

4.  Take the derivative with respect to time.

5.  Substitute the given values and solve.  

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

Air is being inflated into an empty soccer ball at a rate of 21 cubic inches per minute. Find the

rate of change of the radius when the radius is 3 inches.

What do we know!? What's constant? What changes?

What do we want to know!

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

Find if when r = 3

Day 1 intro to related rates notes including group activity.notebook March 10, 2017

HW: p. 146 #13, 16a, 17­19, 25a

Day 1 intro to related rates notes including group activity.notebook March 10, 2017