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# Difference quotient word problem

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1

The height, in metres, of a toy rocket launched at an initial upward velocity of 30 m/s, from a
height  of 1 m, is approximately given by
h=-4.9t^2+30t+1
where t is measured in seconds.
a) Determine the difference quotient for this
function and use it to determine the AROC of
the rocket from t = 4 s to t = 6 s.

b) Determine the IROC of the rocket at t = 5 seconds.

Feb 19, 2020

#1
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a)  We don't need the difference quotient for this one

Average rate of change  =   h(6)   -h(4)

__________

6  -  4

h(6)  = -4.9(6)^2 + 30(6)  + 1  = 4.6

h(4) = -4.9(4)^2 + 30 (4) + 1    = 42.6

So

Avg rate of change  =    [ 4.6 - 42.6]             -38

___________   = _______     =   -19ft / s

6  -   4                  2

b)  Difference Quotient

f ( t + h)  -  f(t)

____________

h

f ( t + h) =  -4.9 ( t + h)^2  + 30 (t + h)  + 1  =    -4.9t^2 - 9.8th - 4.9h^2  + 30 t + 30 h + 1

f ( t)  =  -4.9t^2  + 30t  + 1

So

[ -4.9t^2  - 9.8 th - 4.9h^2 + 30t + 30 h +1   ] -  [ -4.9t^2  +  30t + 1 ]

______________________________________________________   =

h

-9.8 th  + 30h

___________  =

h

h ( -9.8t  + 30)

____________   =      -9.8t  + 30

h

Instantaneous rate of change at 5 sec =    -9.8(5)  + 30  = -19 ft/s   Feb 19, 2020