Understanding Functions and Their Derivatives Graphically

Recall that in lecture we studied the secant lines through a fixed point \( P( c,f(c) ) \) on the graph \( y = f(x) \) and a moving point \( Q (b, f(b)) \), where \( b \neq c\). As we discussed in lecture, the limit of the slopes of these secant lines, as \(b \to c\), is equal to the derivative \(f'(c)\). The tangent line at \(x = c\) is the line which: (i) passes through the point \( (c, f(c)) \); and (ii) has slope \(f'(c)\).

A function appears in the SageMath plot below. Its tangent line at the point \(x = c\) is plotted in red. As you move the slider, the SageMath plot displays the coordinates of the point \((c, f(c))\) and the equation of the tangent line at \(x = c\). Use the SageMath plot to answer the questions in the exercise.

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2. For this exercise, use the SageMath plot below. Set the function — either F1, F2, F3, or F4 — to be the one indicated in your lab report.