Chapter 3: Applications of Differentiation
Section 3.10: Antiderivatives
A particle moving along the x-axis has acceleration x″t=5 t+3, initial velocity x′0=−7, and initial position x0=8. Find xt, its position function.
The antiderivative of the acceleration x″ is the velocity
x′t=5 t22+3 t+c
The additive constant c can be determined to be c=−7 by solving the equation x′0=−7 for c.
The antiderivative of the velocity x′t=52 t2+3 t−7 is the position
xt=52 t33+3 t22−7 t+C
Note the use of C as the second additive constant introduced. This makes it easier to distinguish between the steps of the calculation. Finally, the constant C can be determined to be C=8 by solving the equation x0=8 for C. Hence, the position function is
Tools≻Load Package: Student Calculus 1
Apply the AntiderivativePlot command to the acceleration x″t; include the initial velocity condition.
Apply the AntiderivativePlot command to the velocity x′t; include the initial position condition.
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