1. 2.7 Related Rates

2. Example: Water is draining from a cylindrical tank at 3. liters/second
Example: Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
Find
(We need a formula to relate V and h. )
(r is a constant.)

3. 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.

4. Hot Air Balloon Problem:
Given:
How fast is the balloon rising?
Find

5. Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.
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? B A

6. Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.
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? B A

7. 2.8 Linear approximations and differentials

8. For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.
We call the equation of the tangent the linearization of the function.

9. Linear approximation
Recall the equation of the tangent line of f(x) at point ( a, f(a) ) :
This is called the linear approximation or tangent line approximation of f at a.
The linear function
is called linearization of f at a .
Examples on the board.

10. 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.

11. 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?
error in r
error in A
maximum error in A

12. 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%