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Computational notebooks have emerged as the platform of choice for data science and analytical workflows, enabling rapid iteration and exploration. By keeping intermediate program state in memory and segmenting units of execution into so-called "cells", notebooks allow users to execute their workflows interactively and enjoy particularly tight feedback. However, as cells are added, removed, reordered, and rerun, this hidden intermediate state accumulates in a way that is not necessarily correlated with the notebook's visible code, making execution behavior difficult to reason about, and leading to errors and lack of reproducibility. We present NBSafety, a custom Jupyter kernel that uses runtime tracing and static analysis to automatically manage lineage associated with cell execution and global notebook state. NBSafety detects and prevents errors that users make during unaided notebook interactions, all while preserving the flexibility of existing notebook semantics. We evaluate NBSafety's ability to prevent erroneous interactions by replaying and analyzing 666 real notebook sessions. Of these, NBSafety identified 117 sessions with potential safety errors, and in the remaining 549 sessions, the cells that NBSafety identified as resolving safety issues were more than $7\times$ more likely to be selected by users for re-execution compared to a random baseline, even though the users were not using NBSafety and were therefore not influenced by its suggestions.