## University Calculus: Early Transcendentals (3rd Edition)

$(C)$ represent position $s$, $(A)$ represent velocity $v$ and $(B)$ represent acceleration $a$.
To recognize the curve $(a)$ and its derivative curve $(a')$, what I would do first is to look for corresponding points: change-of-sign points in $(a')$ should correspond to change-of-direction points in $(a)$. The reason is that - As $a'\gt0$, the graph of $a$ moves upward. - As $a'\lt0$, the graph of $a$ moves downward. So a change in the sign of $a'$ will signify a change in the direction of $(a)$ at that exact same point. Here we see that the red curve $(C)$ and the blue curve $(A)$ are such a pair. The first time $(A)$ changes sign from positive to negative, $(C)$ changes from moving upward to downward. The second time $(A)$ changes sign from negative to positive, $(C)$ changes from moving downward to upward. Similarly, the blue curve $(A)$ and the yellow line $(B)$ are another pair. At the point $(B)$ changes sign from negative to positive, $(A)$ changes from moving downward to upward. Therefore, $(C)$ represent position $s$, $(A)$ represent velocity $v$ and $(B)$ represent acceleration $a$.