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This is the solution to the problem. However, I don't understand how they got to the part that says "and the parallel vector is $i+4j$..". What does this mean and how did they derive that?
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$\begingroup$For any real number $m$, the vector $(1,m)$ determines a line of slope $m$ through the origin: simply note that the line through $(0,0)$ and $(1,m)$ has rise $m$ and run $1$.
In this case, your $m=4$, so the vector $(1,4) = 1i + 4j = i+4j$ is a vector of the appropriate slope, hence parallel to the tangent.
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