Download Truck Problem

2.7 Related Rates
Example: Water is draining from a cylindrical tank at 3
liters/second. How fast is the surface dropping?
3
cm
dV
L
 3000
 3
sec
dt
sec
dh
Find
dt
(We need a formula to
relate V and h. )
V   r 2h
dV
2 dh
r
dt
dt
cm3
2 dh
3000
r
sec
dt
(r is a constant.)
cm3
3000
dh
sec

dt
 r2
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 to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.
Hot Air Balloon Problem:
Given:


4
d
rad
 0.14
dt
min
How fast is the balloon rising?
dh
Find
dt
h
tan  
500
d
1 dh
2
sec 

dt 500 dt

1 dh

 sec   0.14  
4
500 dt

h

500ft
 
dh
2  0.14   500 
dt
2
2
ft
dh
140

min dt
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   4
10
1
30   3
10
32  42  z 2
9  16  z 2
25  z 2
5 z
B
z 5
y3
A
x4
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 x d y  z
2
2
2
1
1 dz
dx
dy
40
2 x 10  42 y 30 10
2z  3
dt
dt
dt
32  42  z 2
dz
4  40  3  30
2 5
9  16  z
dt
2 dz
25

z
dz
250  5
50 
dt
dt
5 z
B
z 5
y3
A
x4
miles
50
hour
2.8 Linear approximations
and differentials
For any function f (x), the tangent is a close approximation
of the function for some small distance from the tangent
point.
y
f  x  f  a
We call the equation of the
tangent the linearization of
the function.
0
xa
x
Linear approximation
Recall the equation of the tangent line of f(x) at point ( a, f(a) ) :
y  f  a   f   a  x  a 
This is called the linear approximation or tangent line approximation
of f at a.
The linear function
L  x   f  a   f   a  x  a 
is called linearization of f at a .
Examples on the board.
Differentials
The ideas behind linear approximations are sometimes
formulated in the notation of differentials.
If y=f(x), where f is a differentiable function, then
• the differential dx is an independent variable,
• the differential dy is a dependent variable and is defined in
terms of dx by the equation
dy  f   x  dx
The next example illustrates the use of differentials in
estimating the errors that occur because of approximate
measurements.
Example: The radius of a circle was measured to be 10 ft with a
possible error at most 0.1 ft. What is the maximum error in
using this value of the radius to compute the area of the circle?
A r
dA
dr
 2 r
dx
dx
2
dA  2 r dr
error in
r
error in A
dA  2   10   0.1
dA  2
maximum error in A
Example (cont.)
• Relative error in the area:
dA 2rdr
dr

2
2
A
r
r
that is, twice the relative error in the radius.
• In our case:
dr
0.1
2  2
 0.02
r
10
• This corresponds to percentage error of 2%