2.7 related rates. example: water is draining from a cylindrical tank at 3 liters/second. how fast...
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2.7 Related Rates
Example: Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
L3
sec
dV
dt
3cm3000
sec
Finddh
dt2V r h
2dV dhr
dt dt (r is a constant.)
32cm
3000sec
dhr
dt
3
2
cm3000
secdh
dt r
(We need a formula to relate V and h. )
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
rad0.14
min
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
ft140
min
dh
dt
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
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 dz
x y zdt dt dt
4 40 3 30 5dz
dt
250 5dz
dt 50
dz
dt
miles50
hour
Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.
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
x0 x a
f x f aWe call the equation of the tangent the linearization of the function.
y f a f a x a
L x f a f a x a
is called linearization of f at a .
Recall the equation of the tangent line of f(x) at point ( a, f(a) ) :
Linear approximation
This is called the linear approximation or tangent line approximation of f at a.
The linear function
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
The next example illustrates the use of differentials in
estimating the errors that occur because of approximate
measurements.
dy f x dx
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?
2A r
2 dA r dr
2 dA dr
rdx dx
error in A
error in r
2 10 0.1dA
2dA maximum error in A
Example (cont.)• Relative error in the area:
that is, twice the relative error in the radius.• In our case:
• This corresponds to percentage error of 2%
r
dr
r
rdr
A
dA2
22
02.010
1.022
r
dr