2.3 Product and Quotient Rules: Higher Order Derivatives

 

The Product Rule   

           

 

Do two ways:

 

 

 

 

 

 

 

 

 

 

 

 

 

Find

 

 

 

 

 

 

 

 

The Quotient Rule

           

 

Examples:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Applications:

#48  Demand and Revenue

The manager of a company that produces graphing calculators determines that when x thousand calculators are produced, they will all be sold when the price is   dollars per calculator.

 

a)     At what rate is demand p(x) changing with respect to the level of production x when 3,000 (x=3) calculators are produced?

 

 

 

 

 

 

 

 

 

 

b)     The revenue derived from the sale of x thousand calculators is R(x) = xp(x) thousand dollars.  At what rate is revenue changing when 3,000 calculators are produced?  Is revenue increasing or decreasing at this level of production?

 

 

 

 

 

 

 

 

#52 Bacterial Populations

A bacterial colony is estimated to have a population of   million t hours after the introduction of a toxin.

 

a)  At what rate is the population changing 1 hour after the toxin is introduced (t=1)?  Is the population increasing or decreasing at this time?

 

 

 

 

 

 

 

 

b)     At what time does the population begin to decline?

 

 

 

 

 

 

 

 

Second Derivative  The derivative of the function that is the first derivative.

 

          NOTATION: 

 

One application is for distance as a function of time –

          s(t)   is distance

          s’(t) = v(t)  is velocity

          s’’(t) = v’(t) = a(t)   is acceleration

 

The change in Inflation Rate is a 2nd derivative

 

Polynomials -  2nd derivative is easy – simpler as higher derivative

Example:               

 

 

 

 

 

 

 

But often, 2nd derivative can get messy!

          #44   

 

 

 

 

 

 

 

 

 

Applications with distance function

#62  After t hours of an 8-hour trip, a car has gone  kilometers

a)     Derive a formula expressing the acceleration of the car as a function of time.

 

 

 

b)     At what rate is the velocity of the car changing with respect to time at the end of 6 hours?  Is the velocity increasing or decreasing at this time?

 

 

 

c)     By how much does the velocity of the car actually change during the seventh hour?

Application:  It is projected that t months from now, the average price per unit for goods in a certain sector of the economy will be  dollars.

 

a)  At what rate will the price per unit be increasing with respect to time 5 months from now?

 

 

 

 

b)  At what rate will the rate of price increase be changing with respect to time 5 months from now?

 

 

 

 

 

 

Higher Order Derivatives

 

         

 

 

#66  Find  where