(1 point) In this exercise, we estimate the rate at which the total personal inc

Question

(1 point) In this exercise, we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population in this area was 961400, and the population was increasing at roughly 9200 people per year. The average annual income was 30593 dollars per capita, and this average was increasing at about 1400 dollars per year. Use the Product Rule and these figures to estimate the rate at which the total personal income was rising in the Richmond-Petersburg area in 1999 Rate of increase of total personal income: dollars.

Explanation / Answer

1) 1st we have to assume variables and rates

Let the population be P, and the average annual income per capita be H, with respective annual increase rates of dP/dt and dH/dt respectively.

P*H = $(961,400 * 30,593)
P = 961,400
H = $30,593
dP/dt = 9200 / yr.
dH/dt = $1400 / yr.

(d/dt)(PH)
= P dH/dt + H dP/dt
= 961,400 * $1400 + $30,593 * 9200
= $1,630,727,600 / yr.

2) y=6xcosx

dy/dx= 6(-xsinx+cosx)

dy/dx=m=6(-pisinpi+cospi)=-6

so m= -6   x1=pi, y1=-6pi

y=mx+b

-6pi=-6*pi+b

b= 0

y=-6pi*x answer

3) y= (t^2+2t+3)(6t^2+4)

dy/dt= (t^2+2t+3)(12t)+(6t^2+4)(2t+2)

for t= 3

f '(3) =(3^2+2*3+3)(12*3)+(6*3^2+4)(2*3+2)

=(9+9)(36)+(54+4)(8)

=(18*36)+(58)*8

=648+464=1112 answer

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