Sec 3.4

1
Sec 3.4

• Related Rates Problems
• An Application of the Chain Rule

2
Derivative as Rate of Change

• Recall If

then
the rate of change of y with respect to x.
So, it measures how fast y is changing with
respect to x.
In particular, if t stands for time, then
measures how fast y is changing with time.
3
Examples

• (1) If P denotes the population of a city, then

measures how fast this population is changing
with time.
(2) If s denotes the distance between two cars,
then
measures how fast the distance between the two
cars is changing with time.
(3) If V denotes the volume of water in some
container, then
measures how fast the amount of water in that
container is changing with time.
4
Related Rates Problems

• Situation Some event occurs in time.
• Given How fast something is changing with time.
• To find How fast something else is changing with
  time.

5
Example

• Situation Air is being pumped into a (spherical)
  balloon.
• Given How fast air is being pumped into the
  balloon, say, 50 cm3 per second.
• To find How fast the diameter of the balloon is
  increasing with time.

6
Another Example

• Situation A ladder 5 meter in length, leaned
  against a wall, now starts to slide down.
• Given How fast the top of the ladder is sliding
  down, say, 0.1 meter per second.
• To find How fast the bottom of the ladder is
  sliding away from the wall.

5
7
Yet Another Example

• Situation A man is walking on a sidewalk.
• Given How fast the man is walking, say, 0.2
  meter per second.
• To find How fast the size of his shadow on the
  ground is changing.

8
One more Example

• Situation An airplane is flying in the air,
  passing right over a radar station on the ground.
• Given How fast the airplane is flying, say, 400
  ms-1 .
• To find How fast its distance from the radar
  station is changing.

9
Still One More Example

• Situation Two ships are sailing.
• Given How fast (and in what direction) each ship
  is sailing, say 120 km/h and 160 km/h,
  respectively.
• To find How fast the distance between the ships
  is changing.

10
Heres Another Example

• Situation Water is leaking from a container of
  the shape of an inverted cone.
• Given How fast water is leaking out, say, 100
  cm3 per second.
• To find How fast the water level in the
  container is decreasing.

11
Heres Yet Another Example

• Situation The space shuttle is being launched.
• Given How fast the space shuttle is rising some
  time after it is launched, say, 500 meter per
  second.
• To find How fast one needs to adjust the
  cameras direction to keep the shuttle in sight.

12
One Last Example

• Situation A roller coaster moving along a track
  part of which is shaped like the parabola y
  x2.
• Given The horizontal speed at a certain point on
  the track, say, 3 meter per second.
• To find The vertical speed then (or there).

13
Example 1 Solution
This 50 cm3s-1 is the rate of change of volume of
air inside the balloon.

• Situation Air is being pumped into a (spherical)
  balloon.
• Given How fast air is being pumped into the
  balloon, say, 50 cm3 per second.
• To find How fast the diameter of the balloon is
  increasing with time.

If we let V denote the volume of air inside the
balloon,
w
then dV/dt is 50.
We want the rate of change of the diameter of the
balloon.
If we let w denote the diameter of the balloon,

then we want to know dw/dt.
14
Example 2 Solution

• Situation A ladder 5 meter in length, leaned
  against a wall, now starts to slide down.
• Given How fast the top of the ladder is sliding
  down, say, 0.1 meter per second.
• To find How fast the bottom of the ladder is
  sliding away from the wall.

This 0.1 ms-1 measures the speed of the top of
the ladder.
If we let y denote the vertical distance
between the floor and the top of the ladder,
y
5
then dy/dt is 0.1
x
If we define x as shown,
We want to find the speed of the bottom of the
ladder.
then we want to know dx/dt.
15
Method of Solution A Summary

1. Read and visualize what event is happening, and
   how the event occurs over time.
2. Identify what rate(s) of change is/are given, and
   what rate of change we are to find.
3. Write these rates as the derivatives of suitably
   chosen variables (with respect to time).
4. Find a relationship between these variables.
5. Obtain a relationship between the rates by
   (implicit) differentiation.
6. Find the unknown rate from the given rate(s).

16
Remarks

1. The rate of change of a variable often depends on
   time. At different moments, the rate can be
   different. The question will specify the
   particular moment of time it is interested in. It
   is often phrased in the form when
2. The equation you form to connect the variables
   must not rely on information that is valid for
   only a particular instant of the event. The
   equation has to be valid for the entire duration
   of the event.
3. In Examples 3 onward, I had not displayed all
   necessary information that would allow us to
   solve them. Such information will be supplied as
   we discuss these examples now

17
Example 3 Solution
This is the speed of the man.

• Situation A man is walking on a sidewalk.
• Given How fast the man is walking, say, 0.2
  meter per second.
• To find How fast the size of his shadow on the
  ground is changing.

then dx/dt is 0.2
then we want to find dy/dt .
Extra information we need (always given in
question)
(1) Length of the lamppost, say, 5 m
(2) Height of the man, say, 1.8 m.
If we call this x,
If y denotes the length of his shadow,
18
Example 4 Solution

• Situation An airplane is flying in the air,
  passing right over a radar station on the ground.
• Given How fast the airplane is flying, say, 400
  ms-1 .
• To find How fast its distance from the radar
  station is changing.

speed of the airplane
If we call this distance x,
then dx/dt is 400.
Extra information we need
If we call the distance from the plane to the
radar station y,
(1) The climbing angle, say, 30o
y
(2) The elevation of the airplane when it is
directly above the radar station, say, 5000 meter
then we want to find dy/dt .
(3) The position of the airplane
19
Example 5 Solution

• Situation Two ships are sailing.
• Given How fast (and in what direction) each ship
  is sailing, say 120 km/h and 160 km/h,
  respectively.
• To find How fast the distance between the ships
  is changing.

Call this distance
x
Call this distance
y
z
Call this distance
We want to know
Extra information we need
(1) Original distance separating the ships, say,
200 km.
(2) Angle between their paths, say, 150o.
20
Example 6 Solution

• Situation Water is leaking from a container of
  the shape of an inverted cone.
• Given How fast water is leaking out, say, 100
  cm3 per second.
• To find How fast the water level in the
  container is decreasing.

Top diameter, say, 100 cm.
Height, say, 150 cm.
V
h
Rate of change of volume V of water in
container.
dV/dt is 100.
If we denote the depth of water in the container
by h,
we want to find dh/dt.
Extra information needed
Dimensions of the (conical) container.
21
Example 7 Solution

• Situation The space shuttle is being launched.
• Given How fast the space shuttle is rising some
  time after it is launched, say, 500 meter per
  second.
• To find How fast one needs to adjust the
  cameras direction to keep the shuttle in sight.

This is velocity at a particular moment, say,
when the shuttle reaches 1000 m above the ground.
If we denote the height of the shuttle above the
ground by h,
h
then dh/dt is 500 when h is 1000.
Need what extra information ?
If we use the angle to indicate the
direction in question,
then we want to find at that
instant when h is 1000.
22
Example 8 Solution

• Situation A roller coaster moving along a track
  part of which is shaped like the parabola y
  x2.
• Given The horizontal speed at a certain point on
  the track, say, 3 meter per second.
• To find The vertical speed then (or there).

at the instant when the roller coaster is at
the point, say (2,4).
dx/dt is 3
we want to find dy/dt
at that particular instant.
23
Method of Solution A Summary

1. Read and visualize what event is happening, and
   how the event occurs over time.
2. Identify what rate(s) of change is/are given, and
   what rate of change we are to find.
3. Write these rates as the derivatives of suitably
   chosen variables (with respect to time).
4. Find a relationship between these variables.
5. Obtain a relationship between the rates by
   (implicit) differentiation.
6. Find the unknown rate from the given rate(s).