1. Lesson 2-9 Derivatives as Functions

2. Objectives Understand the difference between differentiability and continuity

3. Vocabulary Differentiable – a function is differentiable at a point, a, if f’(a) exists

4. Differentiable Definition: A function is differentiable at a if f’(a) exists. It is differentiable on an open interval (a, b) if it is differentiable at every number in the interval. Thrm: If f is differentiable at a, then f is continuous at a. Note: The converse is not necessarily true  a function can be continuous at a, but not be differentiable at a (note the corner and vertical tangents, continuous, but not differentiable!) f has a discontinuity at a, if f is not continuous at a. Note the graphs of examples of functions that are not differentiable below: A CornerA DiscontinuityVertical Tangent f(x) = |x| -2x+2 if x < 0 f(x) = 3x if x > 0 f(x) = (x – 1) 0.33333

5. Differentiation Notation We always need to state what we are taking the derivative with respect to. While in this course we only deal with one variable, in more advanced calculus courses we deal with more than one variable! f’(x), dy/dx, df/dx, d( f(x) )/dx, D(f(x)), D x (f(x)) are all good examples of different styles of notation. y’ is not desirable as it doesn’t tell us which variable we are taking the derivative with respect to! y’(x) is good.

6. Example 1 For the function f(x) pictured below, tell whether the statement is true or false. A.f(x) is continuous at 0. B.f(x) is differentiable at 0. C.f(x) is continuous at 2. D.f(x) is differentiable at 2. E.f(x) is continuous at 3. F. f(x) is differentiable at 3. G.f(x) is continuous at 4. H.f(x) is differentiable at 4. T F T T T F F F 12345

7. Graphical Example 2 Given f, draw f’ function derivative

8. Graphical Example 3 Given f, draw f’ function derivative

9. Graphical Example 4 Given f, draw f’ y = e x

10. Graphical Example 5 Given f, draw f’ Cubic square

11. Summary & Homework Summary: –A function that has a derivative at all points is continuous at all points, but not all continuous functions have derivatives for all points –Differentiable implies Continuous, but Continuous may not imply Differentiable –Derivatives are the slope of the tangent line Homework: pg 173 - 175: 4, 5, 12, 19, 21, 24, 37