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A particle is ejected with an initial velocity of 3 m/s , starts moving along a path such that its velocity varies quadratically, governed by 𝑓1(𝑑) = βˆ’π‘‘^2 + 3𝑑 + 𝛼 for the first 1 hour. After 1 hour its velocity varies linearly, governed by 𝑓2(𝑑) = πœ‡π‘‘ + 𝛽. Here, 𝛼,πœ‡ and 𝛽 are constants. With reference to this scenario perform the following: Q1.1 Write the velocity function 𝑓(𝑑) of the particle for all time, where time 𝑑 is in hours. Q1.2 Determine the values of the constants 𝛼,πœ‡ and 𝛽 so that the piecewise velocity function 𝑓(𝑑) defined in Q1.1 satisfies the conditions of Lagrange’s mean value theorem in the time interval 0 ≀ 𝑑 ≀ 2. Q1.3 Determine time instance, 𝑑 = 𝑐 at which the instantaneous velocity of the particle is equal to its average velocity in the time interval 0 ≀ 𝑑 ≀ 2. Q1.4 Obtain equations of the tangent at (𝑐,𝑓(𝑐)) and the chord joining the end points (0,𝑓(0)) and (2,𝑓(2)). Q1.5 Plot the velocity function 𝑓(𝑑) time interval 0 ≀ 𝑑 ≀ 2 and vertical line 𝑑 = 𝑐. Also, plot the tangent line and the chord in the same graph.

A particle is ejected with an initial velocity of 3 m/s , starts moving along a path such that its velocity varies quadratically, governed by 𝑓1(𝑑) = βˆ’π‘‘^2 + 3𝑑 + 𝛼 for the first 1 hour. After 1 hour its velocity varies linearly, governed by  𝑓2(𝑑) = πœ‡π‘‘ + 𝛽. Β Here, 𝛼,πœ‡ and 𝛽 are constants. Β 
With reference to this scenario perform the following:
Q1.1 Write the velocity function 𝑓(𝑑) of the particle for all time, where time 𝑑 is in hours.
Β Q1.2 Determine the values of the constants 𝛼,πœ‡ and 𝛽 so that the piecewise velocityΒ 
function 𝑓(𝑑) defined in Q1.1 satisfies the conditions of Lagrange’s mean valueΒ 
theorem in the time interval 0 ≀ 𝑑 ≀ 2. Β 
Q1.3 Determine time instance, 𝑑 = 𝑐 at which the instantaneous velocity of the particle isΒ 
equal to its average velocity in the time interval 0 ≀ 𝑑 ≀ 2.Β 
Q1.4 Obtain equations of the tangent at (𝑐,𝑓(𝑐)) and the chord joining the end pointsΒ 
(0,𝑓(0)) and (2,𝑓(2)).
Q1.5 Plot the velocity function 𝑓(𝑑) time interval Β 0 ≀ 𝑑 ≀ 2 and vertical line 𝑑 = 𝑐. Β Also,Β 
plot the tangent line and the chord in the same graph.Β 
Β 

Grade:12

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