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# Other applications with rates of change

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1. Section 3.4b Other applications with rates of change

2. 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 its velocity then? After 8 seconds, and its velocity was 0 ft/sec (d) When did the parachute pop out? How fast was the rocket falling then? After about 11 seconds, and it was falling at about 90 ft/sec

3. 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 highest point) (f) When was the rocket’s acceleration greatest? When was the acceleration constant? The acceleration was greatest just before the engine stopped. The acceleration was constant from t = 2 to t = 11, while the rocket was in free fall.

4. Derivatives in Economics In manufacturing, the cost of production c(x) is a function of x, the number of units produced. The marginal cost of production is the rate of change of cost with respect to the level of production, so it is called dc/dx. Ex: Suppose c(x) represents the dollars needed to produce x tons of steel in one week. It costs more to produce x + h tons per week, and the cost difference divided by h is the average cost of producing each additional ton: The average cost of each of the additional h tons produced

5. Derivatives in Economics In manufacturing, the cost of production c(x) is a function of x, the number of units produced. The marginal cost of production is the rate of change of cost with respect to the level of production, so it is called dc/dx. The limit of this ratio as h 0 is the marginal cost of producing more steel per week when the current production is x tons. Marginal cost of Production:

6. Guided Practice Suppose it costs dollars to produce x radiators when 8 to 10 radiators are produced, and that gives the dollar revenue from selling xradiators. Your shop currently produces 10 radiators a day. Find the marginal cost and marginal revenue. The marginal cost of producing one more radiator a day when 10 are being produced is

7. Guided Practice Suppose it costs dollars to produce x radiators when 8 to 10 radiators are produced, and that gives the dollar revenue from selling xradiators. Your shop currently produces 10 radiators a day. Find the marginal cost and marginal revenue. The marginal revenue:

8. Guided Practice When a certain chemical was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while but then stopped growing and began to decline. The size of the population at time t (hours) was Find the growth rates at t = 0, t = 5, and t= 10 hours. Bacteria growth rate: At t = 0: bacteria/hour At t = 5: bacteria/hour At t = 10: bacteria/hour

9. Guided Practice The position of a body at time t seconds is meters. Find the body’s acceleration each time the velocity is zero. Velocity: Acceleration: Find when velocity is zero: Find acceleration at these times: