Determine the average value of a function using definite integrals
I can interpret the average value geometrically as the height of the “perfect rectangle”
I can interpret and write integrals representing average values of functions in context
I can set-up and evaluate an integral expression to determine the average value of a function
Quick Lesson Plan
The concept of the average value of a function is introduced graphically with simple approximations of the area under a given curve. By the end of the lesson, students will have intuitively developed the formula for the average value of any function and will have connected their new formula to its graphical representation.
Students will need minimal intervention or guidance on today’s Activity. The given function is a familiar parabola in the first quadrant. The closed interval [0, 4] allows students to easily see the rectangles they create in question 3 when estimating the mysterious value “c.”
Budget time for student discussion of their responses to Activity question 6. Realizing that the average value of a function is not only the height of our perfect rectangle but also the average of all the y-values on the interval is surprising for most students! Explaining that av(f) is the average of a function’s y-values reinforces their understanding of an integral as a summation (or accumulation) function. Just like finding an average when they were younger, the average value of a function is found by adding together values (integrating) and then dividing (by the quantity b - a).
Prior to the AP Exam, practice the correct notation and linkage when writing average value statements: incorrect placement of division by (b - a) creates linkage errors. AP Readers are instructed to penalize students for this algebraic error.
Using trapezoidal or RAM approximations to estimate an integral before finding its average value is also common on the AP Exam. When interpreting their answer, students should write about “approximate average temperature,” or “approximate average height.”
For excellent AP practice, have students work through the following: 2016 AB5 (which asks for average volume and is useful after Topic 8.9), 2014 AB1 (which asks for average ROC as well as the average value of a function), 2013 AB3 (which we like because it talks about coffee!), and 2011 AB1 (which asks for average velocity --- this is an integral problem, not a slope problem).
Challenges may present themselves when the average value is negative. Areas beneath the x-axis are often problematic, and now we are constructing a rectangle of “negative” height and finding a negative average value. Provide practice for this concept.
Additionally, student notation must be arithmetically correct! A common error occurs when students calculate the value of an integral and equate it to an average value by suddenly including the denominator, (b - a). Interpreting an average value requires more than a brief mention of the value. Unfortunately, students often forget to discuss the interval or include correct units. For example, 2018 AB 4 required the following interpretation: The average height of the tree over the time interval 2 < t < 10 is 263/32 meters.