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