Class 12 Applications Of Derivatives · Case Study4 Markscase_based_application
Class 12 Maths · Applications Of Derivatives · Case Study #30
A hypothetical electric-vehicle count is modeled by V(t)=t³/5−5t²/2+25t−2. Here t=1,2,3,… denotes years 2001,2002,2003,… respectively. Use the continuous polynomial to study monotonicity; negative outputs are not meaningful vehicle counts.
Part (1) 2 Marks
Can this model meaningfully estimate the count in the year 2000? Justify.
Answer: No: it predicts V(0)=−2.
- The year 2000 corresponds to t=0.
- V(0)=−2, which cannot represent a vehicle count.
- The polynomial can be evaluated, but its negative extrapolation is not a meaningful estimate.
Part (2) 2 Marks
Prove that V(t) is strictly increasing.
Answer: V′(t)>0 for all real t.
- Differentiate: V′(t)=(3t²−25t+125)/5.
- The numerator has discriminant 625−1500=−875<0 and positive leading coefficient, so it is positive everywhere.
- Thus V′(t)>0 and V is strictly increasing, in particular on the modeled time range.