Question

The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003.

Date Deaths per day   Date Deaths per day
March 1 0.0079 April 26 0.5620
March 15 0.0638 May 10 0.4630
March 29 0.1944 May 24 0.2897
April 12 0.4435
(a) By using an argument similar to that in the example, estimate the number of people who died of SARS in Singapore between March 1 and May 24, 2003, using both left endpoints and right endpoints. (Round your answers to the nearest whole number.)

left endpoint people
right endpoint people


(b) How would you interpret the number of SARS deaths as an area under a curve?

Answer 1

Answer:

Step-by-step explanation:

check attachment for chart

(a) When a data set is given, the area can be calculated using either left Riemann sum (when we use left end points of each interval) or right Riemann sum (when we use right end points of each interval).

Here the intervals are:

|----------------Interval---------------||----left point--||--right point---|

March 1 - March 15 (t0 to t1) : y0 = 0.0079, y1 = 0.0638

March 15 - March 29 (t1 to t2) : y1 = 0.0638, y2 = 0.1944

March 29 - April 12 (t2 to t3) : y2 = 0.1944, y3 = 0.4435

April 12 - April 26 (t3 to t4) : y3 = 0.4435, y4 = 0.5620

April 26 - May 10 (t4 to t5) : y4 = 0.5620, y5 = 0.4630

May 10 - May 24 (t5 to t6) : y5 = 0.4630, y6 = 0.2897

Hence total no. of deaths using left end point:

$= y0.\bigtriangleup t + y1.\bigtriangleup t + y2.\bigtriangleup t + y3.\bigtriangleup t + y4.\bigtriangleup t + y5.\bigtriangleup t (\bigtriangleup t = 14 days) \\\\=(y0 + y1 + y2+ y3 + y4 +y5).\bigtriangleup t \\\\=(0.0079 + 0.0638 + 0.1944 + 0.4435 + 0.5620 + 0.4630) (14) = 24.2844 \approx 25 deaths (taking next integer value)$

And total no. of deaths using right end point:

$= y1.\bigtriangleup t + y2.\bigtriangleup t + y3.\bigtriangleup t + y4.\bigtriangleup t + y5.\bigtriangleup t + y6.\bigtriangleup t (\bigtriangleup t = 14 days) \\\\=(y1 + y2+ y3 + y4 + y5 + y6).\bigtriangleup t \\\\=(0.0638 + 0.1944 + 0.4435 + 0.5620 + 0.4630 + 0.2897) (14) = 28.2296 \approx 29 deaths (taking next integer value)$

b)  If we are given derivative of a function f(t) as a function of time, that is:

$\frac{\mathrm{d} }{\mathrm{d} t} (f(t)) = f'(t)$

then the function can be evaluated by integrating both the sides. That is:

$f(t) = \int f'(t) dt$

And by definition we know that integration of a curve gives area under the curve. Hence if we have data for SARS deaths per day (i.e. derivative of total deaths as a function of time) then, if this data is plotted on a curve, area under this curve will give us total no. of deaths during a time period.