1. Linearization , Related Rates, and Optimization

2. linearization of f at a
is the standard linear approximation of f at a.
The linearization is the equation of the tangent line, and you can use the old formulas if you like.

3. Ex. Find the linearization of
Solution:
this linear function can be used to
approximate values of x near 0 in the f(x)

4. , then an estimate of G(2.2)
If G(2) = 5 and
using tangent-line approximation is
(A) (B) (C) (D) (E) 13.8

5. Find the linearization of

6. If f is a differentiable function and f (2)=6 and          
, find the approximate value of f (2.1).

7. Keys to Optimization
Steps:
find a primary (what your optimizing)
and secondary equation (concrete info in problem)
2) solve the secondary for one variable
3) substitute it into the primary
4) find extrema of the function
check endpoints and critical #’s
ex. Find two numbers who’s sum is 20 and product is as large
as possible
primary: f(x,y) = xy
f(x)= x(-x+20)
f(x) = -x2+20x
secondary: x + y = 20 =0
y = -x+20
x = 10, so y = 10

8. Find your: primary equation (idea your optimizing)
secondary equation (additional info in problem) .

9. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
There must be a local maximum here, since the endpoints are minimums.

10. Ex.
A farmer plans to fence a rectangular pasture adjacent to a river.
The pasture must contain 180,000 square meters in order to
provide enough grass for the herd. What dimensions would
require the least amount of fencing if none is needed along the
river?

11. Consider rectangles located as shown in the first quadrant and inscribed under a decreasing curve, with the lower left hand corner at the origin and the upper right hand corner on the curve Find the width, height and area of the largest such rectangle.

12. 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.

13. Ex.
Air is being pumped into a balloon at a rate of 6 cubic feet per
minute.
Find the rate of change of the radius when the radius is 2 ft.

14. A pebble gets dropped into a pool and the resulting ripples
are concentric circles that expand from the center.
The radius of the outer ripple is increasing at a constant
rate of 1 foot per second. When the radius is 3 feet,
at what rate is the total area of the water changing?

15. All edges of a cube are expanding at
a rate of 4 inches per second. How fast is the volume
changing when each edge is (a) 1 inch, (b) 3 inches ?

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

17. A ladder 10 ft long rests against a vertical wall.
If the bottom of the ladder slides away from the
wall at a rate of , how fast is the top
of the ladder sliding when the bottom of the
ladder is 6 ft from the wall?
(no –calc)
Label what we know
use pyth. th. to find y
z=10ft
y=8
x= 6 ft
What if we wanted to know the rate of change of the angle with the ground?

18. Group problem: Making Coffee
(# 24 from our book)
Coffee is draining from a conical filter into a cylindrical coffee pot at a rate of
How fast is the coffee in the pot rising when the
coffee in the pot is 5 inches deep?
2) How fast is the level in the cone falling at the moment ?

19. done