lesson 19: related rates
TRANSCRIPT
Lesson 19 (Section 4.1)Related Rates
Math 1a
November 7, 2007
Announcements
I OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
Outline
Introduction
Warmup
Strategy
Examples
Your Turn
Today we’ll look at a direct application of the chain rule toreal-world problems. Examples of these can be found whenever youhave some system or object changing, and you want to measurethe rate of change of something related to it.
Outline
Introduction
Warmup
Strategy
Examples
Your Turn
Problem
Example
An oil slick in the shape of a disk is growing. At a certain time, theradius is 1 km and the volume is growing at the rate of 10,000liters per second. If the slick is always 20 cm deep, how fast is theradius of the disk growing at the same time?
Math 1a - November 07, 2007.GWBWednesday, Nov 7, 2007
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Math 1a - November 07, 2007.GWBWednesday, Nov 7, 2007
Page4of12
A solution
The volume of the disk is
V = 2πr2h.
We are givendV
dt, a certain value of r , and the object is to find
dr
dtat that instant.
Solution
SolutionDifferentiating with respect to time we have
dV
dt= 2πrh
dr
dt=⇒ dr
dt=
1
2πrh· dV
dt.
Now we evaluate:
dr
dt
∣∣∣∣r=1000 m
=1
2π(1 km)(20 cm)· 10, 000 L
s
Converting every length to meters we have
dr
dt
∣∣∣∣r=1 km
=1
2π(1000 km)(0.2 cm)· 10 m3
s=
1
40π
m
s
Outline
Introduction
Warmup
Strategy
Examples
Your Turn
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Strategies for Related Rates Problems
1. Read the problem.
2. Draw a diagram.
3. Introduce notation. Give symbols to all quantities that arefunctions of time (and maybe some constants)
4. Express the given information and the required rate in termsof derivatives
5. Write an equation that relates the various quantities of theproblem. If necessary, use the geometry of the situation toeliminate all but one of the variables.
6. Use the Chain Rule to differentiate both sides with respect tot.
7. Substitute the given information into the resulting equationand solve for the unknown rate.
Outline
Introduction
Warmup
Strategy
Examples
Your Turn
Another one
Example
A man starts walking north at 4 ft/sec from a point P. Five minuteslater a woman starts walking south at 4 ft/sec from a point 500 ftdue east of P. At what rate are the people walking apart 15 minafter the woman starts walking?
Diagram
m
500
ww
500
s
4 ft/sec
4 ft/sec
s =√
(m + w)2 + 5002
15 minutes after the woman starts walking, the woman has traveled(4 ft
sec
)(60 sec
min
)(15 min) = 3600 ft
while the man has traveled(4 ft
sec
)(60 sec
min
)(20 min) = 4800 ft
We want to knowds
dtwhen m = 4800, w = 3600,
dm
dt= 4, and
dw
dt= 4.
Differentiation
We have
ds
dt=
1
2
((m + w)2 + 5002
)−1/2(2)(m + w)
(dm
dt+
dw
dt
)=
m + w
s
(dm
dt+
dw
dt
)At our particular point in time
ds
dt=
672√7081
≈ 7.98587
Outline
Introduction
Warmup
Strategy
Examples
Your Turn
Worksheet