Linearization , Related Rates, and Optimization

The linearization is the equation of the tangent line, and you can use the old formulas if you like.

L x f a f a x a linearization of f at a

f x L x is the standard linear approximation of f at a.

Ex. Find the linearization of

0at x 1)( xxf

Solution: L x f a f a x a

2

1)01(

2

1)0(

)1(2

1)(

1 01)0(

21

21

f

xxf

f

x

xxL

2

11

)0(2

11)(

this linear function can be used to approximate values of x near 0 in the f(x)

If G(2) = 5 and29

10)(

x

xxG

, then an estimate of G(2.2)

using tangent-line approximation is

(A) 5.4 (B) 5.5 (C) 5.8 (D) 8.8 (E) 13.8

Find the linearization of

2 at x cos)(

xxf

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

ex. Find two numbers who’s sum is 20 and product is as largeas possible

Steps:1) find a primary (what your optimizing)

and secondary equation (concrete info in problem)2) solve the secondary for one variable3) substitute it into the primary4) find extrema of the functioncheck endpoints and critical #’s

Keys to Optimization

primary: f(x,y) = xy

secondary: x + y = 20

y = -x+20

f(x)= x(-x+20) f(x) = -x2+20x

202)( xxf =0x = 10, so y = 10

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

.

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?

x x

40 2x

40 2A x x

240 2A x x

40 4A x

0 40 4x

4 40x

10x 40 2l x

w x 10 ftw

20 ftl

There must be a local maximum here, since the endpoints are minimums.

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 wouldrequire the least amount of fencing if none is needed along the river?

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.

328 xy

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.

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.

Ex.

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

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 ?

Hot Air Balloon Problem:

Given:4

rad

0.14min

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

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 topof the ladder sliding when the bottom of the ladder is 6 ft from the wall?

sec4

1 ft

x= 6 ft

sec4

1 ftdt

dx

z=10ftuse pyth. th. to find y

8

106 222

y

y

y=8

sec? ftdt

dy

Label what we know

secft

16

3 falling

or secft

16

3- 0)8(2)

4

12(6)(

change)doesnt ladder of(length 02dt

dx2x

change of rates find toderive 222

dt

dy

dt

dydt

dyy

zyx

(no –calc)

What if we wanted to know the rate of change of the angle with the ground?

Group problem: Making Coffee (# 24 from our book)

Coffee is draining from a conical filter into a cylindrical coffee pot at a rate of

min10

3in

1) 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 ?

done