Reconstructing the position function from a graph of the velocity function: A ca

Question

Reconstructing the position function from a graph of the velocity function: A car is traveling along a marked road which we may consider equivalent to an x-axis. Set X0 = X(0) = 0. Recorded below is a graph of the car velocity V(t) in miles per minute. Find a formula for x(t), for the interval V t 4 minutes, and graph x = x(t). Use the formula for x(t) to determine how for the car traveled in those y minutes. The distance the car traveled is also computable in terms of an area associated to the velocity graph. The reason has to do with the Fundamental Theorem of Calculus. Explain the connection, and verify that this area agrees with your answer from the previous part.

Explanation / Answer

a) v(t) = 1-0.25t for 0<t<2

x(t) = t - 0.125t^2 + C

0 = x(0) = C so C = 0

x(t) = t - 0.125t^2

v(t) = 0.5 for 2<t<4

x(t) = 0.5t + K

x(2) = 2 - .0.125*2^2 = 1.5

1.5 = x(2) = 0.5*2 + K = 1 + K so K = 0.5

therefore:

x(t) = t - 0.125t^2 for 0<t<2 and x(t) = 0.5t+0.5 for 2<t<4

b) x(4) = 0.5*4+0.5 = 2.5: the car went 2.5 miles

c) the area under the graph is the integral of the velocity function, and the position function is also the integral of the velocity function

AREA = 2*(1+0.5)/2 + 2*0.5 = 1.5 + 1 = 2.5

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