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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