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Question
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
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Solution
Given, the estimated of electric vehicles in use at any time t is given by
V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2`
- No, the function cannot be used to calculate the number of vehicles in 2000.
As, t = 1, 2, 3, ... where starting year is 2001, 2002, 2003 ...
Therefore, it could not be used to calculate the year before 2001. - Here, V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2`
`(dV(t))/(dt) = 1/5 xx 3t^2 - 5/2 xx 2t + 25`
V'(t) = `3/5 t^2 - 5t + 25`
For the function to be increasing V'(t) > 0
Here, `3/2 t^2 - 5t + 25 > 0`
Hence, function V(t) > 0
So, it is an increasing function.
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