Zooming In

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

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.

Zooming In A. Graph the function: f (x) = x 3 – 6x x – 4 f (x) = x 3 – 6x x – 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.

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 x – 4 at the point (4, 8).  Since the slope of f (x) at x = 4 equals 11, we write f’ (4) = 11.

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.

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.

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.