Drill Tell whether the limit could be used to define f’(a).
LESSON 3.2 HW: PAGE 114: 2, 4, (EVEN), Differentiability
Objectives Students will be able to find where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents. approximate derivatives numerically and graphically.
Where Might a Derivative Not Exist? At a corner Where the one-sided derivatives differ Example: y = |x|; there is a corner at x = 0 At a cusp Where the slopes of the secant lines approach ∞ from one side and -∞ from the other Example: y = x 2/3, there is a cusp at x = 0
Where Might a Derivative Not Exist? At a vertical tangent Where the slopes of the secant lines approach either ±∞ from both sides Example: f(x) = x 1/3 At a discontinuity Which will cause one or both of the one-sided derivatives be non- existent Example: a step function
Compare the right-hand and left-hand derivatives to show that the function is NOT differentiable.
The graph of a function over a closed interval D is given. At what points does the function appear to be: Differentiable? Continuous but not differentiable? Neither continuous or differentiable? (-3. 2) (2. -2)
The graph of a function over a closed interval D is given. At what points does the function appear to be: Differentiable? Continuous but not differentiable? Neither continuous or differentiable? (-3. 0) (3. 0)
Derivatives on a Calculator For small values of h, the difference quotient is often a good numerical approximation of f’(a). However, using the same value of h will actually yield a much better approximating if we use the symmetric difference quotient: Numerical Derivative at f at some point a: NDER
Example: Computing a Numerical Derivative Compute NDER (x 2, 3). Using h = and
Numerical Derivatives on the TIs MATH nDeriv(function, variable, number)
Example 2 More Numerical Derivatives Compute each numerical derivative.
Theorems Theorem 1: If f has a derivative at x = a, then f is continuous at x = a Theorem 2: If a and b are any two points in an interval on which f’ is differentiable, then f’ takes on every value between f’(a) and f’(b) Example: Does any function have the Unit Step Function as its derivative? No, choose some a = -.5 and some b =.5, Then U(a) = -1 and U(b) = 1, but U does NOT take on any value between -1 and 1.