Experience First

In this lesson your students will need toothpicks to trace the slopes of a function. We keep toothpicks on each table, as they will be needed in many subsequent lessons.

 

Students have been familiarizing themselves with tangent lines and how to find their slope, and in today’s activity we ask the question: is it always possible to draw a tangent line? Students look at two graphs to identify the criteria needed to draw a tangent line. The graph of f(x) is a polynomial where slopes are always defined. Students notice that there is a horizontal tangent at x=-3 and x=2, and more importantly, that the slopes near these x-values are approaching 0. The function acts in predictable ways and there are no sudden changes.

 

In the second part of the activity, students look at a function g(x) that has discontinuities and sharp turns. In question 2a we expect students to identify the discontinuities as places where we can not draw a tangent line, since either the point does not exist or there is a jump in the graph. If a function is not continuous at a point, it doesn’t stand a chance of having a tangent line there.

 

However, this is not the only criteria for drawing a tangent line. In parts b-d, students notice how the slopes change suddenly at x=5. The slope before x=5 is positive 2, but the slope after x=5 is -½. Thus, it is impossible to draw a tangent line at x=5, since we can’t determine the slope right at that point. The same is true for x=3. 

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In question 3, students describe in their own words what criteria is needed to be able to draw a tangent line. In the debrief, we layer on the formal vocabulary of “differentiability” to describe this condition.

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Formalize Later

Looking at graphs and determining discontinuities and sharp turns is an easy access point into differentiability. We save the formal limit notation for the debrief, when we discuss how limits can be used to talk about the slopes coming from both sides of a particular x-value. Point out to students how the limit expressions are similar and different to the ones they have written before (Unit 8). Note that instead of comparing the intended y-values of a function, we are now comparing the intended slopes of a function, aka the values of f’(x).

 

When debriefing question 3, ask students whether it is harder for a function to be differentiable or for a function to be continuous? Have students verbalize that differentiability is a stricter definition than continuity, because it requires continuity AND that there are no sharp turns. We often use an example from Geometry about squares and rectangles to describe the relationship between continuity and differentiability. A square is always a rectangle but a rectangle is not always a square. Similarly, a differentiable function is always continuous (since that is one of the requirements of being differentiable), but a continuous function is not always differentiable (since it could have sharp turns).

 

Questions 2 and 3 on the Check Your Understanding should be familiar to students as they are similar to the continuity justifications we did in Unit 8, but now students must check the extra condition that slopes match from both sides. Students now use two equations to determine the values of the two parameters, a and b. While students may have internalized the process of simply “setting them equal to each other”, continue to encourage mathematical language related to limits. “Setting them equal” is really about evaluating the limits from both sides and seeing if they match. “Setting the slopes equal” is really about evaluating the limits from both sides of the derivative function to see if they match.