The velocity of a particle moving along the t-axis is given by f(t)=2+.05t cm/sec. Use a graph of y=f(t) to find the exact change in position (distance traveled) for the particle from t=2 to t=8.
find f(2) and f(8), then take the difference between them.
To find the exact change in position (distance traveled) for the particle from t=2 to t=8, we need to calculate the definite integral of the velocity function, f(t), over the interval [2, 8].
Step 1: Calculate the anti-derivative of the velocity function, f(t), with respect to t. The anti-derivative of 2+.05t is 2t+0.025t^2.
Step 2: Evaluate the anti-derivative at the upper limit, t=8, and subtract the value at the lower limit, t=2.
(2t+0.025t^2) evaluated from t=2 to t=8 = [(2(8) + 0.025(8)^2) - (2(2) + 0.025(2)^2)]
= [(16 + 0.025(64)) - (4 + 0.025(4))]
= [(16 + 1.6) - (4 + 0.1)]
= [17.6 - 4.1]
= 13.5 cm
Therefore, the exact change in position (distance traveled) for the particle from t=2 to t=8 is 13.5 cm.