Explain the behavior of a function on an interval using the Intermediate Value Theore
I can verify that the conditions for the IVT have been met
I can use the IVT to describe a function’s guaranteed outputs
Quick Lesson Plan
Students use their intuition about real-world scenarios to reason about if certain output values are guaranteed (e.g. Must the odometer read 15 miles?). Each scenario gets at a different condition for the IVT. Students learn about the nuances of justifying a result using a mathematical theorem, one of the big ideas of Calculus.
The formalization of this theorem will take a lot of practice. We recommend you go over each of the Check Your Understanding problems and tomorrow we will be analyzing several student samples to solidify what makes a good justification. We suggest taking time to unpack each part of the theorem and even using a sentence frame to help students get started. An example of such a frame could be:
“Since name of function is continuous on [a,b] and f(a) =___ and f(b)=___, the IVT guarantees that…”
Students should finish the sentence in context, by incorporating the language of the question stem (e.g. “there must be a value r for 1 < r < 3 such that h(r)=-5”). The College Board highlights justification as one of the four key mathematical practices, so it is critical that students know what counts as a proper justification.
The IVT is the first of two existence theorems in Calculus (a theorem claiming that a certain value exists). Later in the year, students will use the Mean Value Theorem and the verbiage required for a formal justification ends up being very similar. Students will revisit the IVT when discussing continuity of derivatives (topic 2.4).
On CYU #4, it is important to mention to students that there are some functions we can assume to be continuous, like sine, cosine, polynomials, and exponential growth and decay functions (as well as their compositions). Thus, the continuity condition has been satisfied and the IVT can be applied.
The scoring guidelines for the FRQs are very particular about the verbiage used when invoking the IVT. The scoring rubric has traditionally awarded 2 points for the IVT, one for finding the y-values of the endpoints and confirming that the desired value is in that interval, and another for invoking the IVT after confirming the condition of continuity (2007 AB3).
Students sometimes confuse the conditions and conclusions of the IVT. They finish their justification by concluding that the function is continuous, when in fact this was just the entry point into the theorem. They also conflate the x-values and y-values in this theorem, so be sure to emphasize when to use a and b, and when to use f(a) and f(b). A quick sketch of the given information is very valuable.