Calculus Notes 3.9 & 3.10: Related Rates & Linear Approximations & Differentials.
Start up:
1. If one side of a rectangle, a, is increasing at a rate of 3 inches per minute while
the other side, b, is decreasing at a rate of 3 inches per minute, which of the
following must be true about the area A of the triangle?
(A) A is always increasing
Answer:
(B) A is always decreasing
1. D
(C) A is decreasing only when a<b
(D) A is decreasing only when a>b
(E) A is constant
2. What is the difference between the function L(x) defined in the text and the
equation of the tangent line y=f(a)+f’(a)(x-a)?
Answer:
2. There is no difference between L(x) and y = f(a) + f ‘ (x) ( x — a ). Since L(x) is
the linear approximation of f at a which uses L(x)= f(a) + f ‘ (x) ( x — a ) to
make that approximation.
Calculus Notes 3.9 & 3.10: Related Rates & Linear Approximations & Differentials.
Start up:
1. If one side of a rectangle, a, is increasing at a rate of 3 inches per minute while
the other side, b, is decreasing at a rate of 3 inches per minute, which of the
following must be true about the area A of the triangle?
(A) A is always increasing
Example 1: Let’s look at why D for #1.
(B) A is always decreasing
What do we know about
(C) A is decreasing only when a<b
triangles and area?
(D) A is decreasing only when a>b
a
(E) A is constant
1
A  ab
b
2
We have some rates given: Rate of a changing: 3 in/min and Rate of b changing: -3
in/min. Where do we get these from? Take the derivative of the area with respect to time.
dA 1  da
db  Since time is not a variable each derivative needs to
  b  a  indicate that. Hence the **/dt in each.
dt 2  dt
dt 
Plug in what we know and simplify.
da
db
 3
 3
dA
3b  3a
dA 1
dt
dt

  b  3   a  3   
dt
2
dt 2
So when is dA/dt decreasing?
When a>b of course.
when
dA
0
dt
Calculus Notes 3.9: Related Rates.
Strategy:
1. Read the problem carefully.
2. Draw a diagram if possible.
3. Introduce notation. Assign symbols to all quantities that are functions of time.
4. Express the given information and the required rate in terms of derivatives.
5. Write an equation that relates the various quantities of the problem. If
necessary, use the geometry of the situation to eliminate one of the variables by
substitution.
6. Use the Chain Rule to differentiate both sides of the equation with respect to t.
7. Substitute the given information into the resulting equation and solve for the
unknown rate.
Calculus Notes 3.9: Related Rates.
Example 2: At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35
km/h and ship B is sailing north at 25 km/h. At what rate is the distance between the
ships changing at 4:00 PM?
3. Introduce notation. Assign
z
1. Read the problem carefully.
symbols to all quantities that
y
2. Draw a diagram if possible.
are functions of time.
4. Express the given information and the dx
required rate in terms of derivatives. dt  35kph
5. Write an equation that relates the
various quantities of the problem.
If necessary, use the geometry of
the situation to eliminate one of
the variables by substitution.
Want
dy
 25kph
dt
dz
when t=4h.
dt
z 2   150  x   y 2
2
x
150-x
Calculus Notes 3.9: Related Rates.
Example 2: At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35
km/h and ship B is sailing north at 25 km/h. At what rate is the distance between the
ships changing at 4:00 PM?
dx
z
dy
dz

35
kph
 25kph
Want
when t=4h.
y
dt
dt
dt
6. Use the Chain Rule to differentiate both
sides of the equation with respect to t.
x
z 2   150  x   y 2
2
150-x
dz
dx
dy
 2 150  x  1  2 y
dt
dt
dt
dx
dy
150

x

1

y



dz
dt
dt

dt
z
2z
7. Substitute the given information
into the resulting equation and
solve for the unknown rate.
dz  150   140    1 35    100  25 

dt
10100


dz  10  35    2500 

dt
10100

dz

dt


2150
10100

 21.4kph
Calculus Notes 3.10: Linear Approximations and Differentials.
Example 3: Atmospheric pressure P decreases as altitude h increases. At a
temperature of 15C, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at
h=1 km, and 74.9 kPa at h=2 km. Use a linear approximation to estimate the
atmospheric pressure at an altitude of 3 km.
Use L(x) equation for the linear approximation. L x  f a  f ' a x  a
In this case it will be:
   
P  h  P  a   P '  a  h  a 
Figure out P ‘ (2):
 
P  1  P  2 
87.1  74.9 12.2
P ' 2 
 

 12.2kPa / km
1 2
1 2
1
Now plug everything in and solve:
P  3   P  2   P '  2  3  2 
P  3    74.9    12.2  1
P  3   62.7 kPa

Calculus Notes 3.10: Linear Approximations and Differentials.
Example 4: The radius of a circular disk is given as 24 cm with a maximum error in
measurement of 0.2 cm.
(a) Use differentials to estimate the maximum error in the calculated area of the disk.
(b) What is the relative error? What is the percentage error?
1.
2.
3.
4.
5.
Read the problem carefully.
Draw a diagram if possible.
Introduce notation. Assign symbols to all quantities that are functions of time.
Express the given information and the required rate in terms of derivatives.
Write an equation that relates the various quantities of the problem. If necessary, use
the geometry of the situation to eliminate one of the variables by substitution.
24cm
A   r2
r  24
dr  0.2
6. Use the Chain Rule to
differentiate both sides of the
equation with respect to t.
dA  2 r  dr 
 

dA  2 24 0.2
7. Substitute the given
information into the
2
resulting equation and
dA

9.6


30
cm
solve for the unknown rate.
So maximum possible error in the calculated area of the disk is about 30cm2.
Calculus Notes 3.10: Linear Approximations and Differentials.
Example 4: The radius of a circular disk is given as 24 cm with a maximum error in
measurement of 0.2 cm.
(a) Use differentials to estimate the maximum error in the calculated area of the disk.
(b) What is the relative error? What is the percentage error?
(b)
Relative error, which is computed by dividing the error by the
total (in this case Area).
A dA
2 r dr 2  24  0.2  0.4
1





2
2
A
A
r
24
60
  24
Percentage error, turn the relative
error from decimal to percent.
 0.016
 1.6%
PS 3.9 pg.202 #2, 7, 8, 11, 12, 19, 27, 31, 32 (9)
PS 3.10 pg.210 #3, 4, 5, 7, 15, 16, 17, 19, 20, 21, 22, 41, 44, 47 (14)
Review pg. 214 #1-90 every 3rd problem (30)