To determine
To explain: The meaning of N′(t)
Explanation
Given:
N(t) is the number in thousands of minimally invasive cosmetic surgery procedures performed in the U.S. for various years.
Explanation:
N′(t) is the rate with which the number of minimally invasive cosmetic surgery procedures is changing with respect to time which are performed in the united states
The unit of N(t) is in thousands.
Thus, unit of N′(t) is thousands of surgeries per year.
To determine
To construct: The table of the estimated values of N′(t).
Answer
The table of the estimated values of N′(t) will be:
t2000200220042006200820102012N′(t)−301.5492.51060.25865.75605.75534.5737
Explanation
Formula used: For small h,
limh→0N(t+h)−N(t)h≈N(t+h)−N(t)h
Calculation:
The value of N′(2000) is computed as follows,
Substitute h=2 in the formula,
N′(2000)=N(2000+2)−N(2000)2=N(2002)−N(2000)2=4897−55002=−301.5
Thus, N′(2000)=−301.5
The value of N′(2002) is computed as follows,
Substitute h=−2 in the formula,
N′(2002)=N(2000)−N(2002)−2=5500−4897−2=−301.5
Substitute h=2 in the formula,
N′(2002)≈N(2004)−N(2002)2=7470−48972=1286.5
Take the average of above two calculations of N′(2002)
N′(2002)=12(1286.5−301.5)=492.5
Thus, N′(2002)=492.5.
The value of N′(2004) is computed as follows,
Substitute h=−2 in the formula
N′(2004)=N(2002)−N(2004)−2=4897−7470−2=1286.5
Substitute h=2 in the formula,
N′(2004)≈N(2006)−N(2004)2=9138−74702=834
Take the average of above two calculations of N′(2004)
N′(2004)=12(1286.5+834)=1060.25
Thus, N′(2004)=1060.25.
Calculation of N′(2006) is as follows,
Substitute h=−2 in the formula,
N′(2006)=N(2004)−N(2006)−2=7470−9138−2=834
Substitute h=2 in the formula,
N′(2006)≈N(2008)−N(2006)2=10897−91382=879.5
Take the average of above two calculations of N′(2006)
N′(2006)=12(879.5+834)=856.75
Thus, N′(2006)=856.75.
Calculation of N′(2008) is as follows,
Substitute h=−2 in the formula,
N′(2008)=N(2006)−N(2008)−2=9138−10897−2=879.5
Substitute h=2 in the formula,
N′(2008)≈N(2010)−N(2008)2=11561−108972=332
Take the average of above two calculations of N′(2008)
N′(2008)=12(879.5+332)=605.75
Thus, N′(2008)=605.75.
Calculation of N′(2010) is as follows,
Substitute h=−2 in the formula,
N′(2010)=N(2008)−N(2010)−2=10897−11561−2=332
Substitute h=2 in the formula,
N′(2010)≈N(2012)−N(2010)2=13035−115612=737
Take the average of above two calculations of N′(2010),
N′(2010)=12(332+737)=534.5
Thus, N′(2010)=534.5.
Calculation of N′(2012) is as follows,
Substitute h=−2 in the formula,
N′(2012)=N(2010)−N(2012)−2=11561−13035−2=737
Thus, N′(2012)=737.
Thus, the required table will be:
t2000200220042006200820102012N′(t)−301.5492.51060.25865.75605.75534.5737
To determine
To sketch: The graph of N and N′
Explanation
By hypothesis,
t2000200220042006200820102012N(t)5500489774709138108971156113035
From part (b),
t2000200220042006200820102012N′(t)−301.5492.51060.25865.75605.75534.5737
Graph:
Use an online calculator to plot the N(t) on vertical axis and t on horizontal axis.
Use an online calculator to plot the N′(t) on vertical axis and t on horizontal axis.
To determine
To explain: The process to get more accurate values for N′(t).
Explanation
Since limh→0N(t+h)−N(t)h≈N(t+h)−N(t)h, the less time interval will be, the more accurate will be the data of N′(t).
