Calc H Unit 2Day 6 – Continuity / Differentiability AND “Derivation of the Slope Function”

Differentiability of functions: (Or, asked another way: “When/where is a function NOT differentiable?”)

Derivation of the Slope Function:

The purpose of this “lab” is to see if we can come up with any kind of pattern regarding

a function f(x) and its “slope function” m(x).

A few questions first:

1)What is “local linearity”?

2.) What is meant by “instantaneous rate of change”?

3.) On a graph, what gives us the IROC at a point?

4.) When IROC is negative, how would you describe the graph?

5.) When IROC is positive, how would you describe the graph?

Directions: PLEASE FOLLOW CAREFULLY!!

  1. Set your graphing window to [-9.4, 9.4] for the x-values and [-20, 20] for the y-values.

Graph the function you have been assigned.

  1. Carefully transfer your graph to graph paper. Use the trace function to generate points so your graph can be transferred as accurately as possible. (Your graph should look “taller” than the calculator’s since you will be using a square scale and your calculator is not.)
  1. Because of local linearity, we can say that a curve has a slope at any given point. We will find the slope of the curve at the x-values given in Chart #1 and then later GRAPH these values as a separate function, the slope function.
  1. To complete the table, first recall that when we discuss the slope of the curve at a point, we really mean the slope of the line tangent to the curve at that point. There are several ways to figure out this slope. We are going to use a function of our calculator called DRAW. (2nd function on PRGM key.)
  1. Press DRAW, then choose Option 5: “Tangent”.
  1. Enter the first x-value on the chart and press ENTER. (If there is already a number in for x, just enter a new number anyway and it will override it.)

The calculator will draw the tangent line and give you the equation in slope intercept form. (Enter both the equation and the slope on the chart.)

  1. Repeat for each of the x-values and complete the chart.
  1. Notice that the slope changes as the x-value changes!!

Therefore, the slope DEPENDS on the x-value, so it is a FUNCTION of x.

(We will say Slope = m(x).) Transfer the values of the slope into the m(x) t-chart.

  1. Use a different colored pen or pencil to graph m(x) on the same graph as your original f(x).

Clearly label the f(x) and m(x) graphs.

  1. Using the points on m(x), find the equation for m(x). Show your work for this under #10.
  1. Now go back and use the limit definition to find the DERIVATIVE OF f(x).

Show this work under #11.

What do you notice?

LAB – DERIVATION OF THE SLOPE FUNCTION

CHARTS, DATA, AND COMPUTATIONS

Group 1:Group 2:

Group 3:Group 4:

x / Eqn. of tangent line from
calculator () / Slope (m) / x / m(x)
(recopy the slope column!)
-6 / -6
-5 / -5
-4 / -4
-3 / -3
-2 / -2
-1 / -1
0 / 0
1 / 1
2 / 2
3 / 3
4 / 4
5 / 5
6 / 6

Show your work for #10 & #11 below.

10.

11.