A worksheet for derivative calculation:

Lab 3 Part I, Calculus I

Unity College, Carrie Diaz Eaton

Consider an environmental indicator which can be described by the following equation:

P(t)=et

where t is in years.

1. Sketch a graph of P(t) and from that, sketch what you expect P’(t) to look like.

2. Write an equation for P(t+∆t)

3. Try to calculate the derivative algebraically using the limit definition of derivative

4. Create a spreadsheet that calculates the derivative numerically at 21 different points of the graph:  t = 0, .5, 1, 1.5, … 10

(Hint: Create a table for with 21 rows (t = 0, .5, 1, 1.5, … 10) and 8 columns: t,P(t),  then for∆t=1 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t, then for∆t=.1 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t, and for∆t=.01 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t.)

5. Results Graph a scatterplot of P(t) and your estimates of P’(t) based on the numerical work performed in #4.

6. Discussion Do the visual results based on visual, algebraic and/or numerical work agree?

7. Discussion Contrast the initial and long-term value of the instantaneous rate of change.  

Discussion Suppose a different function was hypothesized for P(t), such as t^2 or e^(-t).  How would you modify your spreadsheet to approximate the graph of the derivative?