x f(x) g(x) f’(x) g’(x)

3 1 8 -3 -5

6 3 -2 4 5

8 -1 3 4

1 2 -6 5 0

For each expression below, use the table above to find the value of the derivative at x = 3

3

1

g x

f x

f x g x f g x

5 19, , 5

48 81

Calculus Warm-up

Olympic National Park, WashingtonGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007

2.6 Related Rates 2014

Olympic National Park, WashingtonGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007

2.6 Related Rates

Olympic National Park, WashingtonGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007

2.6 Related Rates

We have seen how the Chain Rule can be used to find dy/dx implicitly. Another important use of the Chain Rule is to find the rates of change of two or more related variables that are changing with respect to time.

Examples: •The rate of change in the radius of a balloon being inflated.

•The rate of movement of a rotating spotlight moving across a wall.

•The approach rate of two vehicles moving towards an intersection.

Steps for Related Rates Problems:

1. Draw a picture (sketch).

2. Write down known information.

3. Write down what you are looking for.

4. Write an equation that relates the variables.

5. Differentiate both sides with respect to time.

6. Solve for one of the rates.

Three Important Rules for Related Rates Problems:

1. Use Geometry to establish the relationship between the variables.

3. Wait until after you have differentiated to substitute any values for the variables.

2. Only changing quantities get variables.

A simple one:

Find: dy/dt when x=1, given that dx/dt =2 when x=1.

3If 3y x

Consider a sphere of radius 10cm. (Possibly a soap bubble or a balloon.) Suppose that the radius of the sphere is changing at an instantaneous rate of 0.1 cm/sec. At what rate is the sphere changing when the radius is 10 cm.?

34

3V r

24dV dr

rdt dt

2 cm4 10cm 0.1

sec

dV

dt

3cm

40sec

dV

dt

The sphere is growing at a rate of . whenthe radius is 10 cm.

340 cm / sec

SphereProblem:

A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of the outer ripple isIncreasing at a constant rate of 1 foot per second. When theradius is 4 feet, at what rate is the total area of the disturbedwater changing?

2A r

Given rate: 1dr

dt

Find: when r = 4dA

dt

The area is changing at a rate of whenthe radius is 4.

28 ft / sec

Ripples inThe Pond:

Do work

Water is draining from a cylindrical tank at a constant rate of 3000 cubic centimeters/second. How fast is the surface dropping?

dV

dt

3cm3000

sec

Finddh

dt2V r h

2dV dhr

dt dt (r is a constant.)

32cm

3000sec

dhrdt

3

2 2

cm3000

seccm

dh

dt r

(We need a formula to relate V and h. )

CylindricalTank Problem:

2

3000/ seccm

r

Hot Air Balloon Problem:

Given:

4

rad0.14

min

d

dt

How fast is the balloon rising atthe instant when ?

Find when 4

dh

dt

tan500

h

2 1sec

500

d dh

dt dt

2

1sec 0.14

4 500

dh

dt

h

500ft

Hot Air Balloon Problem:

Given:4

rad

0.14min

d

dt

How fast is the balloon rising?

Finddh

dt

tan500

h

2 1sec

500

d dh

dt dt

2

1sec 0.14

4 500

dh

dt

h

500ft

2

2 0.14 500dh

dt

1

12

4

sec 24

ft140

min

dh

dt

The formula for the volume of a cone is

Find the rate of change of the volume if is 2 inches

per minute and h = 3r when r = 6 inches.

21.

3V r hdr

dt

2

2

3

3

3 6 2

216 / min

V r r

V

V in

2

2

3

1

31

33

V r h

V r r

V r

Cube Problem:

All edges of a cube are expanding at a rate of 3 cm. per second. How fast is the volume changing when each edge is

a) 1 cm. and b) 10cm.?

3

3

)9 / sec

)900 / sec

a cm

b cm

BC Homework 2.6 pg.154

13-23 odd,31,33,35,43

AB Homework 2.6 pg.154

13-23 odd

AB Homework 2.6 Day 2pg.154 27,31,33,35,43,45

RELATED RATES – DAY 2

4x

3y

B

A

5z

Truck Problem:Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.

How fast is the distance between the trucks changing 6 minutes later?

r t d 1

40 410

130 3

10

2 2 23 4 z 29 16 z

225 z5 z

4x

3y

30dy

dt

40dx

dt

B

A

5z

Truck Problem:

How fast is the distance between the trucks changing 6 minutes later?

r t d 1

40 410

130 3

10

2 2 23 4 z 29 16 z

225 z5 z

2 2 2x y z

2 2 2dx dy dzx y zdt dt dt

4 40 3 30 5dz

dt

250 5dz

dt

50dz

dt

miles50

hour

Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.

Lamppost Problem:

A man 6 feet tall walks at a rate of 5 feet per second towards a lamppost that is 20 feet above the ground. When he is 10 feet from the base of the light,

a)At what rate is the tip of his shadow moving?

b) At what rate is the length of his shadow changing?

50/ sec.

7ft

15/ sec.

7ft

y

y x

yx

2.6 Related Rates

is decreasing at a rate of 40

A plane is flying over a radar

0 mph 10

tracking station. If

when , mi

what is the s

of the p p need la e?

s s

2 2 36

2 2 0

2 10 400

2

s x

ds dxs xdt dt

dx

dt x

2 2 2

?

10 6

8

x

x

x

500 mphdx

dt

400ds

dt

10s

?dx

dt

500 mphspeed

2=50

rate of change in the of el

Given , ( in feet and in seconds),

find the of the

camera at

evation

10 seco after n lds ift-off.

h th t t

h

2=50h t t

?d

dt

10 =5000h

tan2000

h

2

2

2

1sec

20001 1

cos 2000

2

c

0

o

0 0

s

d dh

dt dtd dh

dt dt

d

d

d

dtt

h

x

2 22000 5000x

2 2

10

2000cos

2000 5000

100 10 1000t

dh

dt

2 radians per second

29

2.6 Related Rates (#27)

A 25 ft. ladder is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 ft. per second.

a)How fast is the top of the ladder moving down the wall when its base is 7 ft.?

a)Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall

b)Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

Homework

• MMM pgs. 68-73