Theorem 1 Theorem 1 The chain ruleThe chain rule

If k(x) =If k(x) =ff (g(x)), then (g(x)), then

k’(x)k’(x) = = f ’f ’ ( ( g(x)g(x) ) ) g’(x)g’(x)

If k(x) =If k(x) =ff (g(x)), then (g(x)), thenk’(x)k’(x) = = f ’f ’ (( g(x)g(x) )) g’(x)g’(x)k(x) = ( xk(x) = ( x22 + x) + x)33

k’(x)k’(x) = = 3 3 ( x( x2 2 + x)+ x)22(2x +1) (2x +1)

The chain ruleThe chain ruleIf y = If y = ((uu))33 and u(x) = x and u(x) = x22 + x + x

then dy/dx = dy/du du/dxthen dy/dx = dy/du du/dx

dy/du = 3(dy/du = 3(uu))22 d duu/dx = /dx = 2x + 12x + 1

dy/dx = dy/dx = 3(3(uu))22 (2x + 1)(2x + 1)

= = 3(3(xx22 + x + x))22 (2x + 1)(2x + 1)

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/dx =

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/du dy/dx = dy/du

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/du du/dxdy/dx = dy/du du/dx dy/dx = dy/dx =

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/du du/dxdy/dx = dy/du du/dx dy/dx = 4/3 udy/dx = 4/3 u1/31/3

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/du du/dxdy/dx = dy/du du/dx dy/dx = 4/3 udy/dx = 4/3 u1/31/3(6x) but can’t quit(6x) but can’t quit

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/du du/dxdy/dx = dy/du du/dx dy/dx = 4/3 udy/dx = 4/3 u1/31/3(6x) but can’t quit(6x) but can’t quit dy/dx = 8xdy/dx = 8x

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/du du/dxdy/dx = dy/du du/dx dy/dx = 4/3 udy/dx = 4/3 u1/31/3(6x) but can’t quit(6x) but can’t quit dy/dx = 8x(3xdy/dx = 8x(3x22-1)-1)1/31/3

#49 y=u#49 y=u4/34/3 u = 3x u = 3x22-1-1

dy/dx = dy/du du/dxdy/dx = dy/du du/dx dy/dx = 4/3 udy/dx = 4/3 u1/31/3(6x) but can’t quit(6x) but can’t quit dy/dx = 8x(3xdy/dx = 8x(3x22-1)-1)1/31/3

or rewrite the problem y=(3xor rewrite the problem y=(3x22--1)1)4/34/3

y’ = 4/3(3xy’ = 4/3(3x22-1)-1)1/31/3(6x)(6x)

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xwhere x is the number of housing where x is the number of housing

starts.starts.

The number of starts in the next t The number of starts in the next t months is x(t) = million months is x(t) = million units per year.units per year.

How many starts this month? How many starts this month?

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xwhere x is the number of housing where x is the number of housing

starts.starts.

The number of starts in the next t The number of starts in the next t months is x(t) = million units months is x(t) = million units per year.per year.

How many starts this month? 70/55 How many starts this month? 70/55 millionmillion

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xwhere x is the number of housing starts.where x is the number of housing starts.

The number of starts in the next t months is The number of starts in the next t months is x(t) = million units per year.x(t) = million units per year.

How many starts this month? 70/55 millionHow many starts this month? 70/55 million

How many jobs created? 1.42(70/55) millionHow many jobs created? 1.42(70/55) million

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xwhere x is the number of housing where x is the number of housing

starts.starts.

The number of starts in the next t The number of starts in the next t months is x(t) = million months is x(t) = million units per year.units per year.

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xwhere x is the number of housing where x is the number of housing

starts.starts.

The number of starts in the next t The number of starts in the next t months is x(t) = million units months is x(t) = million units per year.per year.

Find dN/dt, the rate of created jobs t Find dN/dt, the rate of created jobs t months from now.months from now.

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dxdN/dt = dN/dx

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dx dx/dt = dN/dt = dN/dx dx/dt =

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dx dx/dt = 1.42dN/dt = dN/dx dx/dt = 1.42

2

2

7 140 700

3 80 550

t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dx dx/dt dN/dt = dN/dx dx/dt

= 1.42= 1.42

2

2

7 140 700

3 80 550

t t

t t

2 2(3 80 550)(14 140) (7 140 7000)(6 80)t t t t t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dx dx/dt dN/dt = dN/dx dx/dt

= 1.42= 1.42

Find the rate of growth of N, one year Find the rate of growth of N, one year from now.from now.

2

2

7 140 700

3 80 550

t t

t t

2 2

2 2

(3 80 550)(14 140) (7 140 700)(6 80)

(3 80 550)

t t t t t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dx dx/dt dN/dt = dN/dx dx/dt

= 1.42= 1.42

Set t = Set t =

2

2

7 140 700

3 80 550

t t

t t

2 2

2 2

(3 80 550)(14 140) (7 140 7000)(6 80)

(3 80 550)

t t t t t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dx dx/dt dN/dt = dN/dx dx/dt

= 1.42= 1.42

Set t = 12Set t = 12

2

2

7 140 700

3 80 550

t t

t t

2 2

2 2

(3 80 550)(14 140) (7 140 7000)(6 80)

(3 80 550)

t t t t t t

t t

#81 Number of #81 Number of construction jobs created construction jobs created is N(x) = 1.42xis N(x) = 1.42xx(t) = million units per year.x(t) = million units per year.

dN/dt = dN/dx dx/dt dN/dt = dN/dx dx/dt

= 1.42= 1.42

2

2

7 140 700

3 80 550

t t

t t

2 2

2 2

(3 80 550)(14 140) (7 140 7000)(6 80)

(3 80 550)

t t t t t t

t t

EconomicsEconomics

The total daily cost of manufacturing The total daily cost of manufacturing dryers is given by dryers is given by

C(x) = 6000 + 100 x - 0.2 xC(x) = 6000 + 100 x - 0.2 x22..

C(x) = 6000 + 100 x - 0.2 C(x) = 6000 + 100 x - 0.2 xx22 on [0, 350] on [0, 350]

Find the actual cost of Find the actual cost of producing the 101 st dryer.producing the 101 st dryer.

C(101) – C(100) = C(101) – C(100) =

C(x) = 6000 + 100 x - 0.2 C(x) = 6000 + 100 x - 0.2 xx22..on [0, 350]on [0, 350]

Find the actual cost of producing the Find the actual cost of producing the 101 st dryer.101 st dryer.

C(101) – C(100) = 14059.80 – C(101) – C(100) = 14059.80 – 14000.0014000.00

= 59.80= 59.80

C(x) = 6000 + 100 x - 0.2 C(x) = 6000 + 100 x - 0.2 xx22 on [0, 350] on [0, 350]

Find the marginal cost. C’(x) =Find the marginal cost. C’(x) =

C’(x) = 100 – 0.40 xC’(x) = 100 – 0.40 x

Evaluate C’(100) and interpretEvaluate C’(100) and interpret

C’(100) = 100 – 40.00 = $60.00 C’(100) = 100 – 40.00 = $60.00

C(x) = 6000 + 100 x - 0.2 C(x) = 6000 + 100 x - 0.2 xx22 on [0, 350] on [0, 350]

C(101)-C(100) / 1 = 59.80C(101)-C(100) / 1 = 59.80

C’(100) = 60C’(100) = 60

C’(100) = C’(100) =

C’(100) is approximated with h = 1C’(100) is approximated with h = 10

(100 ) (100)limh

C h C

h

C(x) = 6000 + 100 x - 0.2 C(x) = 6000 + 100 x - 0.2 xx22 on [0, 350] on [0, 350]

C’(100) = 60C’(100) = 60

InterpretInterpret

The cost of producing the 101 st dryer The cost of producing the 101 st dryer will be approximately $60.00will be approximately $60.00

C’(x) = 100 – 0.40 xC’(x) = 100 – 0.40 x

C’(50) = 100 – 20 = 80C’(50) = 100 – 20 = 80

InterpretInterpret

The cost of producing the 51 st dryer The cost of producing the 51 st dryer will be approximately $80.00will be approximately $80.00

C’(x) = 100 – 0.40 x Find C’(x) = 100 – 0.40 x Find the approximate cost of the approximate cost of producing the 201producing the 201stst dryer.dryer.

C’(x) = 100 – 0.40 x Find C’(x) = 100 – 0.40 x Find the approximate cost of the approximate cost of producing the 201producing the 201stst dryer.dryer.

20.020.0

4.14.1

DVD PlayersDVD Players

C(x) = C(x) = 0.0001 x0.0001 x33–0.08 x–0.08 x22 +40 x +40 x +5000+5000

Daily total cost is given aboveDaily total cost is given above

Find the marginal cost functionFind the marginal cost function

C’(x) =C’(x) = 20.0003x 20.0003 0.16x x20.0003 0.16 40x x

On [0, 500] C(x) = On [0, 500] C(x) = .0001 x .0001 x3 3 – – 0.08 x0.08 x22 +40 x +5000 +40 x +5000

C’(x) =C’(x) =

Find the marginal cost for producing Find the marginal cost for producing 200, 300, 400, 500 DVD players200, 300, 400, 500 DVD players

C’(200) = 20, C’(300) = 19,C’(200) = 20, C’(300) = 19,

C’(400) = 24, C’(600) = 52C’(400) = 24, C’(600) = 52

20.0003 0.16 40x x

C’(200) = 20, C’(300) = 19,C’(200) = 20, C’(300) = 19,C’(400) = 24, C’(600) = 52C’(400) = 24, C’(600) = 52

What did it cost to produce the 201What did it cost to produce the 201stst DVD player? DVD player?

301301stst ? 401 ? 401stst ? 601 ? 601stst ? ?

C’(x) =C’(x) =Cost of producing 101Cost of producing 101stst is is

20.0003 0.16 40x x

20.0003(100) 3

C’(x) =C’(x) =Cost of producing 101Cost of producing 101stst is is

27.0027.00

1.11.1

20.0003 0.16 40x x

20.0003(100) 3

Average CostAverage Cost

If 100 items cost $300, what is If 100 items cost $300, what is the average cost? $ 300 / 100 =the average cost? $ 300 / 100 =

$3$3

Average CostAverage Cost

If 100 items cost $300, what is If 100 items cost $300, what is the average cost? $ 300 / 100 =the average cost? $ 300 / 100 =

$3$3 If x items cost $ C(x), what is the If x items cost $ C(x), what is the

average cost?average cost? Answer is Answer is

__ ( )C xC x

x

C(x) = 400 + 20xC(x) = 400 + 20x

Find the average cost function.Find the average cost function.

__ ( )C xC x

x 1400 20 400 20

400 20x x

xx x x

Evaluate the average Evaluate the average cost in the long run.cost in the long run.

Evaluate Evaluate 400

lim 20x x

Evaluate Evaluate

20.020.0

0.10.1

400lim 20x x

= $20= $20

Which is expected because the fixed Which is expected because the fixed cost remains constant while it is cost remains constant while it is spread over more and more product.spread over more and more product.

400lim 20x x

= 400 x= 400 x-1-1 + 20 + 20

Find the marginal average cost Find the marginal average cost function.function.

__

C x

__2'( ) 400C x x

C(x) = 0.0001 xC(x) = 0.0001 x3 3 – 0.08 – 0.08 xx22 +40 x +5000 +40 x +5000 Find the average cost for Find the average cost for

producing DVD players.producing DVD players. Average cost = (x)= Average cost = (x)= 0.0001 x0.0001 x2 2 – 0.08 x +40 – 0.08 x +40

+5000x+5000x-1-1

__

C

(x) = 0.0001 x(x) = 0.0001 x2 2 – 0.08 x – 0.08 x +40 +5000/x, find +40 +5000/x, find

__

(100)C

__

C

(x) = 0.0001 x(x) = 0.0001 x2 2 – 0.08 x – 0.08 x +40 +5000/x, find +40 +5000/x, find

83.083.0

0.10.1

__

(100)C

__

C

= 0.0001 x= 0.0001 x2 2 – 0.08 x – 0.08 x +40 +5000 x+40 +5000 x-1-1, find , find (x)(x)A.A. 0.02x – 0.08 + 40 + 5000 x 0.02x – 0.08 + 40 + 5000 x -2-2

B.B. 0.002x – 0.08 + 40 + 5000 x 0.002x – 0.08 + 40 + 5000 x -2-2

C.C. 0.0002x – 0.08 + 5000 x 0.0002x – 0.08 + 5000 x -2-2

D.D. 0.0002x – 0.08 - 5000 x 0.0002x – 0.08 - 5000 x -2-2

__

'C

__

( )C x

= 0.0001 x= 0.0001 x2 2 – 0.08 x – 0.08 x +40 +5000 x+40 +5000 x-1-1, find , find (x)(x)A.A. 0.02x – 0.08 + 40 + 5000 x 0.02x – 0.08 + 40 + 5000 x -2-2

B.B. 0.002x – 0.08 + 40 + 5000 x 0.002x – 0.08 + 40 + 5000 x -2-2

C.C. 0.0002x – 0.08 + 5000 x 0.0002x – 0.08 + 5000 x -2-2

D.D. 0.0002x – 0.08 - 5000 x 0.0002x – 0.08 - 5000 x -2-2

__

'C

__

( )C x

= = 0.0002x – 0.08 - 5000 x0.0002x – 0.08 - 5000 x--

22, find (500), find (500)

A.A. 11

B.B. 22

C.C. 33

D.D. 00

__

'C

__

'( )C x

= = 0.0002x – 0.08 - 5000 x0.0002x – 0.08 - 5000 x--

22, find (500), find (500)

A.A. 11

B.B. 22

C.C. 33

D.D. 00

__

'C

__

'( )C x

= = 0.0002x – 0.08 - 5000 x0.0002x – 0.08 - 5000 x--

22, find (500), find (500)

A.A. 11

B.B. 22

C.C. 33

D.D. 00

__

'C

__

'( )C x

= = 0.0002x – 0.08 - 5000 x0.0002x – 0.08 - 5000 x--

22, find (500), find (500)

A.A. 11

B.B. 22

C.C. 33

D.D. 00

__

'C

__

'( )C x

The price is $500The price is $500

Find the revenue if you sell 1000.Find the revenue if you sell 1000. R(x) = x p(x) = 1000 (500)R(x) = x p(x) = 1000 (500) = $500,000= $500,000

LoudspeakersLoudspeakers

The price function is p(x) The price function is p(x) = -0.02 x + 400 on [0, = -0.02 x + 400 on [0, 20000]20000] Find the revenue function.Find the revenue function. R(x) = x p(x) =R(x) = x p(x) = -0.02 x-0.02 x22 + 400 x + 400 x

R(x) = x p(x) = -0.02 xR(x) = x p(x) = -0.02 x22 + 400 + 400 x x

Find the marginal revenue Find the marginal revenue function.function.

R’(x) =R’(x) =-0.04 x + -0.04 x + 400

R’(x) = -0.04 x + 400 R’(x) = -0.04 x + 400 How much revenue for 2001How much revenue for 2001stst one?one?

A.A. 320320

B.B. 300300

C.C. 280280

R’(x) = -0.04 x + 400 R’(x) = -0.04 x + 400 How much revenue for 2001How much revenue for 2001stst one?one?

A.A. 320320

B.B. 300300

C.C. 280280

R(x) = -0.02 xR(x) = -0.02 x22 + 400 x + 400 x

Suppose the cost function is Suppose the cost function is C(x) = 100x + 200,000 for the C(x) = 100x + 200,000 for the

loudspeakersloudspeakers Find the profit function.Find the profit function.

Find the profit Find the profit function.function. R(x) = -0.02 xR(x) = -0.02 x22 + 400 x + 400 x C(x) = 100x + 200,000 C(x) = 100x + 200,000 P(x) = R(x) – C(x) = P(x) = R(x) – C(x) =

-0.02 x-0.02 x22 + 400 x – + 400 x – ((100x + 200,000 100x + 200,000 ))

P(x) = -0.02 xP(x) = -0.02 x22 + 300 x + 300 x -- 200000 200000

P(x) = -0.02 xP(x) = -0.02 x22 + 300 x + 300 x -- 200000 200000 Find the marginal profit functionFind the marginal profit function

P’(2000) = profit for the P’(2000) = profit for the salesale

of the 2001 loudspeaker of the 2001 loudspeaker

'( ) 0.04P x x'( ) 0.04 300P x x $220

Sketch the graph of Sketch the graph of P(x)P(x)

P(x) = -0.02 xP(x) = -0.02 x22 + 300 x + 300 x –– 200000 Find 200000 Find P’(1000)P’(1000)

P(x) = -0.02 xP(x) = -0.02 x22 + 300 x + 300 x –– 200000 Find 200000 Find P’(1000)P’(1000)

260.0260.0

0.10.1

P’(1000) = 260 P’(1000) = 260 InterpretInterpret

P’(1000) = 260 P’(1000) = 260 InterpretInterpret

Upon building 1001 dreyers, the Upon building 1001 dreyers, the profit on the sale of the 1001profit on the sale of the 1001stst dryer is $260. dryer is $260.