Shown below is the graph \(y=f(x)\text{.}\) If we choose a point \(Q\) on the graph to the left of the \(y\)-axis, is the slope of the secant line through \(P\) and \(Q\) positive or negative? If we choose a point \(Q\) on the graph to the right of the \(y\)-axis, is the slope of the secant line through \(P\) and \(Q\) positive or negative?
2
Shown below is the graph \(y=f(x)\text{.}\)
If we want the slope of the secant line through \(P\) and \(Q\) to increase, should we slide \(Q\) closer to \(P\text{,}\) or further away?
Which is larger, the slope of the tangent line at \(P\text{,}\) or the slope of the secant line through \(P\) and \(Q\text{?}\)
3
Group the functions below into collections whose secant lines from \(x=-2\) to \(x=2\) all have the same slopes.
Exercises — Stage 2
4
Give your best approximation of the slope of the tangent line to the graph below at the point \(x=5\text{.}\)
5
On the graph below, sketch the tangent line to \(y=f(x)\) at \(P\text{.}\) Then, find two points \(Q\) and \(R\) on the graph so that the secant line through \(Q\) and \(R\) has the same slope as the tangent line at \(P\text{.}\)
6
Mark the points where the curve shown below has a tangent line with slope \(0\text{.}\)
(Later on, we'll learn how these points tell us a lot about the shape of a graph.)
