Calculating Instantaneous Rate of Change (Lesson 9.2 Day 1)
Unit 9 - Day 3
Understand that limits turn an estimate of the instantaneous rate of change into the exact value of the slope
Set-up and evaluate a limit expression that gives the slope at a single point
Write the equation of a tangent line to a curve at a given point
Quick Lesson Plan
Based on the common myth about a penny dropped from the Empire State Building being able to kill someone (and the subsequent Mythbuster episode to disprove it), students explore the actual speed of a penny in freefall. On page 1, students calculate the penny’s average speed over a 1.5 second interval and then estimate the penny’s actual speed after 1.5 seconds. The too-low and too-high estimates garnered great discussion. While some used way too high and way too low estimates, many groups calculated slopes between t-values just to the left and just to the right of t=1.5 and concluded that the true slope would be somewhere in between. Many were able to estimate the penny’s speed as being 48 ft per second based on these measurements! As you are monitoring students, record what strategies you see being used, and determine how you want to sequence these strategies in the debrief.
On page 2, students actually calculate the slope on various intervals that get smaller and smaller. We introduce the notation of “a” and “h” and ask students to interpret what h means. Students realize that as the two x-values get closer and closer together, the slope calculated in the last column becomes closer and closer to the penny’s true speed at t=1.5. The formal limit notation and evaluation is done as a class in the debrief.
Note that we always use “a” when calculating slope at a specific point and “x” when we are finding the slope at any point, i.e. the derivative function. We have found that this alleviates much of the confusion around the notation. The biggest hang-up for students tends to be the shift away from the (y2-y1)/(x2-x1) definition of slope to the (f(a+h)-f(a))/h definition. The debrief of the table given in question 4 should make this connection clear for students. Ask the students why the denominator of the slope column doesn’t have a difference in it. Students should be able to explain that h already represents the difference between the two x-values. If students are still confused by the idea of h, ask them how they would calculate x2-x1 in their table. It should become evident that a+h - a=h. We make this explicit again in the important ideas, where we mark the interval between the two x-values as h.
To find the equation of the tangent line, we prefer using point-slope form, even though students continue to favor slope-intercept. We allow students to write the equation in either form (all equivalent answers are accepted) but we hope that over time they will see the value in not having to solve for the y-intercept!
Algebra errors abound when students evaluate the limit definition of slope at a point. Proper use of parentheses and a review of how to evaluate functions at expressions like “a+h” is helpful in setting students up to tackle the Check Your Understanding questions. Keep in mind that while the algebraic skills are important, we shouldn’t let students lose sight of what this limit expression represents. More purposeful practice will take place tomorrow!