254e08…: “Of a maroon fish”

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Let $F:\R ^2\to \R$ be a differentiable function that is roughly described by the following table of values:

Let $\vecl (t)$ be the vector-valued function describing the line running from the point $(2,1)$ to $(-2,5)$ as $t$ runs from $0$ to $1$ . Use the table above to estimate $\grad {F}(0,3)$ and use this to estimate $\dd {t}F(\vecl (t))$ at $t = 1/2$ . $\vecl (t) = \vector {\answer {2-4t},\answer {1+4t}}$ $\grad F(0,3) \approx \vector {\answer {-5},\answer {-2.5}}$ $\eval {\dd {t} F(\vecl (t))}_{t=1/2} \approx \answer {10}$