DA with first principles and limits

Here is an example of how I have changed the definition of an exam.  Before, when assessing first principles and limits in my calculus class, I would have each student write an exam.  This year, I am giving students the option of writing an exam or completing the following to demonstrate their learning.

Unit 1 Assessment Project

1.      Create a formula for the distance of an object at any time t.  The formula must be at least degree 2 with a minimum of 2 terms.  Using your formula you must:

·         Determine average velocity of the object for the first 10 seconds.

·         Determine, with an explanation, the instantaneous velocity after 10 seconds.

·         Determine, with an explanation, the instantaneous velocity for any time t.

·         Determine if the object is ever traveling 50 m/s.

2.      Create a piecewise function over the domain of the real numbers. The function must contain at least 3 different pieces and may not be continuous and may not have a limit at at least one point.  Using your function you must:

·         Provide a graph of your function.

·         Explain why your function is not continuous at the specific point.

·         Explain why a limit does not exist at the point above. Explain how you could change your function such that there is a limit.  Calculate the left and right side limits at this point.

3.      Demonstrate, through a real life application, how you can calculate the instantaneous rate of change at a specific data point.  You may use data from a chart, create a video, etc., but the data may not follow a specific function.  The point at which you calculate the rate is entirely up to you, but must have an explanation.

4.      Create three functions, one which:

·          approaches infinite,

·         one which approaches 0,

·         and another which approaches a line which is not 0

as you extend these functions to infinite.  Explain how you can determine the limit as x approaches infinite, for each of your functions.

5.      Create a real life example of a convergent series, and calculate the sum of this series.