OTHER APPLICATIONS WITH RATES OF CHANGE

Section 3.4b

The “Do Now” – p.130-131, #16

(a) How fast was the rocket climbing when the engine stopped?

v = 190 ft/sec

(b) For how many seconds did the engine burn?

2 seconds

(c) When did the rocket reach its highest point? What was itsvelocity then?

After 8 seconds, and its velocity was 0 ft/sec

(d) When did the parachute pop out? How fast was the rocketfalling then?

After about 11 seconds, and it was falling atabout 90 ft/sec

The “Do Now” – p.130-131, #16

(e) How long did the rocket fall before the parachute opened?

About 3 seconds (from the rocket’s highestpoint)

(f) When was the rocket’s acceleration greatest? When wasthe acceleration constant?

The acceleration was greatest just before theengine stopped. The acceleration wasconstant from t = 2 to t = 11, while the rocketwas in free fall.

Derivatives in EconomicsIn manufacturing, the cost of production c(x) is a function ofx, the number of units produced. The marginal cost ofproduction is the rate of change of cost with respect to thelevel of production, so it is called dc/dx.

Ex: Suppose c(x) represents the dollars needed to produce xtons of steel in one week. It costs more to produce x + h tonsper week, and the cost difference divided by h is the averagecost of producing each additional ton:

c x h c x

h

The average cost of each of

the additional h tons produced

Derivatives in EconomicsIn manufacturing, the cost of production c(x) is a function ofx, the number of units produced. The marginal cost ofproduction is the rate of change of cost with respect to thelevel of production, so it is called dc/dx.

The limit of this ratio as h 0 is the marginal cost ofproducing more steel per week when the current productionis x tons.

0

limh

c x h c xdc

dx h

Marginal cost of Production:

Guided PracticeSuppose it costs 3 26 15c x x x x

dollars to produce x radiators when 8 to 10 radiators areproduced, and that 3 23 12r x x x x

gives the dollar revenue from selling x radiators. Your shopcurrently produces 10 radiators a day. Find the marginal costand marginal revenue.

The marginal cost of producing one more radiator a day when10 are being produced is 10c

3 26 15d

c x x x xdx

23 12 15x x

10 3 100 12 10 15c $195

Guided PracticeSuppose it costs 3 26 15c x x x x

dollars to produce x radiators when 8 to 10 radiators areproduced, and that 3 23 12r x x x x

gives the dollar revenue from selling x radiators. Your shopcurrently produces 10 radiators a day. Find the marginal costand marginal revenue.

The marginal revenue:

3 23 12d

r x x x xdx

23 6 12x x

10 3 100 6 10 12r $252

Guided PracticeWhen a certain chemical was added to a nutrient broth inwhich bacteria were growing, the bacterium populationcontinued to grow for a while but then stopped growing andbegan to decline. The size of the population at time t (hours)was 6 4 3 210 10 10b t t t Find the growth rates at t = 0, t = 5, and t = 10 hours.

Bacteria growth rate: 10,000 2000b t t At t = 0: 0 10,000b bacteria/hour

At t = 5: 5 0b bacteria/hour

At t = 10: 10 10,000b bacteria/hour

Guided PracticeThe position of a body at time t seconds ismeters. Find the body’s acceleration each time the velocityis zero.

3 26 9s t t t

Velocity:

Find when velocity is zero:

v t s t

23 4 3 0t t

23 12 9t t Acceleration: a t v t s t 6 12t

23 12 9 0t t

1,3t 3 1 3 0t t

Find acceleration at these times:

21 6m seca

23 6m seca