calculus 12 Limits and Continuity
Specific Curriculum Outcomes
B1 Calculate and interpret average and instantaneous rate of change.
B2 Calculate limits for function values and apply limit properties with and without technology
C1 Identify the intervals upon which a given function is continuous and understanding the meaning of a continuous function
B3 Remove removable discontinuities by extending or modifying a function
B4 Apply the properties of algebraic combinations and composites of continuous functions
A1 Apply, understand and explain average and instantaneous rates of change an extend these concepts to secant lines and tangent line slopes
C2 Understand the development of the slope of a tangent line from the slope of a secant line
C3 Find the equations of the tangent and normal lines at a given point
B1 Calculate and interpret average and instantaneous rate of change.
B2 Calculate limits for function values and apply limit properties with and without technology
C1 Identify the intervals upon which a given function is continuous and understanding the meaning of a continuous function
B3 Remove removable discontinuities by extending or modifying a function
B4 Apply the properties of algebraic combinations and composites of continuous functions
A1 Apply, understand and explain average and instantaneous rates of change an extend these concepts to secant lines and tangent line slopes
C2 Understand the development of the slope of a tangent line from the slope of a secant line
C3 Find the equations of the tangent and normal lines at a given point
activities
 An Introduction to Limits  Hand out a quarter sheet of card stock to each student and ask them to draw and cut out an curvy amoeba shape (you can give it a face and a name if you'd like). Trace this shape onto 1.5 cm graph paper and count squares to estimate the area of this shape. Repeat the process on 1.25 cm, 1 cm, 0.75 cm and 0.5 cm graph paper. Graph these values and talk about what would happen if you continued to repeat this process. This process is called a limit and is used for lots of math. You might discuss Archimedes' estimate of the value of pi using the area of polygons and the idea of limits.
 Limits and Continuity Desmos Activity by Bryn Humberstone  In this activity, students consider left and right limits—as well as function values—in order to develop an informal and introductory understanding of continuity.
 An Intuitive Introduction to Limits  Some nice ideas here for a discussion of what limits are before jumping into calculations.

 Limits and Continuity Crafts from Jonathan Claydon  Jonathan asks his students to design a piecewise function that demonstrates aspects of the limits and continuity. The function has to have five continuity problems (left/right disagreement, removable discontinuity, and unbounded), they had to demonstrate they could find the limit at various points on their function, and they had to explain the situation.

 Limit Definition of the Derivative from Shawn Cornally  Begin with the speedometer. How do you know how fast you’re going? Invariably kids come up with the idea that you measure how long it took you to go a certain distance. I then pose the question: “Well how do you know how fast you’re going right now, at this very second?” After that I let the kids loose with office chairs.

 Introducing Tangents with a Car Speedometer  Using a video of a car trip to determine the average speed of the trip. How does this speed relate the the instantaneous speed of the carÉ
 Hilbert's Curve, and the usefulness of infinite results in a finite world video (17min) from 3Blue1Brown  An interesting discussion of the Hilbert Curve space filling fractal, continuity and limits to infinity.
