1. Zooming In

2. Objectives  To define the slope of a function at a point by zooming in on that point.  To see examples where the slope is not defined.  ES: Explicitly assessing information and drawing conclusions

3. Zooming In  Most functions we see in calculus have the property that if we pick a point on the graph of the function and zoom in, we will see a straight line.

4. Zooming In A. Graph the function: f (x) = x 3 – 6x 2 + 11x – 4 f (x) = x 3 – 6x 2 + 11x – 4 B. Zoom in on the point (4, 8) until the graph looks like a straight line. C. Pick a point on the curve other than the point (4, 8) and estimate the coordinates of this point. D. Calculate the slope of the line through these two points.

5. Zooming In  The slope of a function is called its derivative, and is denoted f’ (x).  The number we just computed is an approximation for the slope or derivative of f (x) = x 3 – 6x 2 + 11x – 4 at the point (4, 8).  Since the slope of f (x) at x = 4 equals 11, we write f’ (4) = 11.

6. Zooming In  Local linearity is a property of differentiable functions that says that if you zoom in on a point on the graph of the function, the graph will eventually look like a straight line with a slope equal to the derivative of the function at the point.  A function is differentiable at a point if its derivative exists at that point.

7. Zooming In  Not every function has a derivative at all of its points.  Graph the function f (x) = |x| and zoom in at the point (0, 0).  Notice that f’ (0) does not exist, because as we zoom in on (0, 0) the graph does not look like a straight line.

8. Conclusion  The slope of a function is called its derivative.  Local linearity is a property of differentiable functions.  Not every function has a derivative at all of its points.