AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.6: Related Rates.

AP CALCULUS AB

Chapter 4:Applications of Derivatives

Section 4.6:Related Rates

What you’ll learn about Related Rate Equations Solution Strategy Simulating Related Motion

…and whyRelated rate problems are at the heart of

Newtonian mechanics; it was essentially to solve such problems that calculus was invented.

What are Related Rate Equations?

Any equation involving two or more variables that are differentiable

functions of time “t” can be used to find an equation that relates their

corresponding rates.

We use implicit differentiation to differentiate several variables with

respect to time.

Strategy for Solving Related Rate Problems To solve related rates problems:

1. Draw a picture.

2. Identify a variable whose rate of change you seek.

Strategy for Solving Related Rate Problems Solving related rates problems (cont)3. Find a formula relating the variable whose rate

of change you seek with one or more variables whose rate of change you know. This is the hard part. The formulas can come from geometry, physical laws, or wherever. Sometimes the AP test will supply the formula, but don’t count on it. It is important to keep variables variable. Never plug in a number which can “freeze the picture” until after differentiating.

Strategy for Solving Related Rate Problems Solving related rates problems (cont)4. Differentiate implicitly with respect to time t.

Remember all those rules (product, quotient, chain, etc.). It is tempting to forget them when dealing with variables like r and h, which do not look like functions of time.

5. Plug and chug. It is safe now to “freeze the picture.”

6. Make sure that you have answered the problem. Write out your answer in a sentence (with units) to see that it makes sense.

Example: Finding Related Rate Equationsa) Assume that the radius r of a sphere is a differentiable

function of t and let V be the volume of the sphere. Find an equation that relates dV/dt and dr/dt.

V = dV/dt =

b) Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt.

V = dV/dt =

You try: Given x2 + y2 = z2. Find an equation that relates dx/dt, dy/dt, and dz/dt.

Example 1 – Related Rates Ex 1: 1. Draw a picture.


dy/dt=-2 ft/sec

x

y

dx/dt=?

25 ft

dt

dx

Example 1 (cont’d) – Related Rates3. Find a formula relating the variable whose rate of

change you seek with one or more variables whose rate of change you know.

4. Differentiate implicitly with respect to time t.

222 25 yx

022 dt

dyy

dt

dxx

Example 1 (cont’d) – Related Rates5. Plug and chug (Freeze the picture)

20

400

625225

2515

2

2

222

x

x

x

x

ft/sec 5.12

3

40

60

6040

06040

02152202

022

dt

dxdt

dxdt

dxdt

dxdt

dyy

dt

dxx

Example 2 – Related Rates Ex 2: 1. Draw a picture.


Leaking out at a rate of 5 cubic cm/secSo dV/dt=-5

9 cmh

3 cm

r

At what rate was the height of water changing when h=4 cm ?

dt

dh

Example 2 (cont’d) – Related Rates3. Find a formula relating the variable whose rate of

change you seek with one or more variables whose rate of change you know.

3

3

1

9

3

hr

h

r

3

2

2

2

27

1

93

1

33

1)(

3

1

hhV

hh

hV

hh

hV

hrV

Example 2 (cont’d) – Related Rates4. Differentiate implicitly with respect to time t.

dt

dhh

dt

dV

dt

dhh

dt

dV

hhV

2

2

3

9

327

27

1)(

Example 2 (cont’d) – Related Rates5. Plug and chug (Freeze the picture)

cm/sec 895.0

16

95

9

165

49

5

9

2

2

dt

dh

dt

dhdt

dhdt

dhdt

dhh

dt

dV

Example 2: A hot-air balloon rising straight up from a level field is tracked by a range finder 500’ from the lift off point. At the moment the range finder’s elevation angle is π/4, the angle is increasing at the rate of 0.14 radians per minutes. How fast is the balloon rising at that moment?

1. Identify variables

2. Draw picture – label

3. Find formula

4. Differentiate implicitly

5. Substitute explicit values into differentiated formula

6. Interpret solution in a sentence

Example 3 - A Highway Chase A police cruiser, approaching a right-angled intersection from the north, is chasing

a speeding car that has turned the corner and is now moving straight east. When the

cruiser is 0.8 mi north of the intersection and the car is 0.6 mi to the east, the police

determine with radar that the distance between them and the car is increasing at

15 mph. If the cruiser is moving at 60 mph at the instant of measurement, what

is the speed of the car?

Example 3 A police cruiser, approaching a right-angles intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car?



3. Find formula




Example 3 - A Highway Chase (cont’d) Let the distance of the speeding car from the intersection.

Let the distance of the police car from the intersection.

Let the distance between the two cars.

We know that / 15, / 60,

x

y

z

dz dt dy dt

2 2 2

0.6, 0.8, and 1.

Use the known information and the Pythagorean Theorem to find / .

2 2 2

2 0.6 2 0.8 60 2 1 15

105 The speeding car is travelling at 105 mp

x y z

dx dt

d d dx dy dzx y z x y z

dt dt dt dt dtdx

dtdx

dt

h.

You try Exercise 13 An Airplane is flying at an altitude of 7 mi and passes directly over a radar antenna as shown in the figure on p 251. When the plane is 10 mi from the antenna (s = 10), the radar detects that the distance s is changing at the rate of 300 mph. What is the speed of the airplane at that moment?

1 & 2 Identify variables / Draw picture – label

3. Find formula




Example 4 Water runs into a conical tank at the rate of 9 ft3 /min. The tank stands point down and has a height of 10’ and a base radius of 5’. How fast is the water level rising when the water is 6’ deep?



3. Find formula



6. Interpret solution in a sentence(note 2nd solution strategy on p 249)

What are related rate equations and what can they tell us?

You TubeJust Math Tutoring

Related Rate Equations Examples 1 & 2

In Review: Strategy for Solving Related Rate Problems

In particular, identify the variable whose rate of change

you seek and the variable (or variables) whose rate of change you know.

1. Understand the Problem

2. Develop a Mathematical Model of the Problem Draw pictures (many of these

problems involve geometric figures) and label the parts that are important to the

problem. Be sure to distinguish constant quantities from variables that change over

time. Only constant quantities can be assigned numerical values at the start.

Th

3. Write an equation relating the variable whose rate of change you seek with

the variable(s) whose rate of change you know. e formula is often geometric,

but it could come from a scientific application.

Be sure to follow all the differentiation rules

4. Differentiate both sides of the equation implicitly with respect to time.

. The Chain Rule will be especially

critical, as you will be differentiating with respect to the parameter .t

In Review: Strategy for Solving Related Rate Problem Notice that it is only

safe to do this after the differentiation step. Substituting too soon "freezes the

picture" and makes changeable var

5. Substitute values for any quantities that depend on time.

iables behave like constants, with zero derivatives.

Translate your mathematical result into the problem setting

(with appropriate units) and decide whether the result makes s

6. Interpret the Solution

ense.

AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.6: Related Rates.

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