1. Section 2.6 Differentiability

2. Local Linearity Local linearity is the idea that if we look at any point on a smooth curve closely enough, it will look like a straight line Thus the slope of the curve at that point is the same as the slope of the tangent line at that point Let’s take a look at this idea graphically

9. Once we have zoomed in enough, the graph looks linear! Thus we can represent the slope of the curve at that point with a tangent line!

10. The tangent line and the curve are almost identical! Let’s zoom back out

12. Differentiability We need that local linearity to be able to calculate the instantaneous rate of change –When we can, we say the function is differentiable Let’s take a look at places where a function is not differentiable

13. Consider the graph of f(x) = |x| Is it continuous at x = 0? Is it differentiable at x = 0? –Let’s zoom in at 0

17. No matter how close we zoom in, the graph never looks linear at x = 0 –Therefore there is no tangent line there so it is not differentiable at x = 0 We can also demonstrate this using the difference quotient

18. Definition The function f is differentiable at x if exists Thus the graph of f has a non-vertical tangent line at x We have 3 major cases –The function is not continuous at the point –The graph has a sharp corner at the point –The graph has a vertical tangent

19. Example

20. Note: This is a graph of It has a vertical tangent at x = 0 –Let’s see why it is not differentiable at 0 using our power rule Example

21. Is the following function differentiable everywhere? Graph What values of a and b make the following function continuous and differentiable everywhere? Example

22. 13) A cable is made of an insulating material in the shape of a long, thin cylinder of radius r 0. It has electric charge distributed evenly throughout it. The electric field, E, at a distance r from the center of the cable is given by Is E continuous at r = r 0 ? Is E differentiable at r = r 0 ? Sketch a graph of E as a function of r.