Homework Quiz Page 105 Exercises 2 & 8 Show work!

3.2 Differentiability What you will learn about today How f’(a) might fail to exist Differentiability implies Local Linearity Derivatives on a calculator – BEWARE! Differentiability implies continuity Intermediate Value Theorem for derivatives Why? Graphs of differentiable functions can be approximated by their tangent lines at points where the derivative exists.

When does a function fail to have a derivative? A function will not have a derivative at a point P(a, f(a)) where the slopes of the secant lines fail to approach a limit as x approaches a. This might look like a Corner, Cusp, or Vertical Tangent line.

Or – a biggie - Discontinuity Consider the function U(x) = -1, x < 0 Graph 1, x ≥ 0 Find the right and left hand limits.

Example 1: Finding where a function is Not Differentiable Find all points in the domain of f(x)=|x - 2| + 3 where f is not differentiable. Think graphically first. You try exercise 1

Most functions ARE Differentiable! Differentiable functions include Polynomials Rational functions Trigonometric functions Exponential functions Logarithmic functions Compositions of the above functions *Our job this chapter is to learn to find derivatives of all these types of functions!

Differentiability implies Local Linearity Let’s do exploration 1 to “see” local linearity! Graph f(x) and g(x) f(x) = |x| + 1 g(x) = √(x ) Let’s ZOOM IN to see the differences!

Derivatives on a Calculator To evaluate at a point x = a Math #8 nderiv(function, x, a) to compute the derivative at x = a Y= nderiv (function, x, x) to graph the derivative function. You can then evaluate the derivative on a point on the graph of f(x) Graph f(x), CALC, dy / dx, trace or type in a, get derivative

BEWARE!!!! You need to understand where a derivative does not exist, your calculator can not tell! Calculator uses the “symmetric difference quotient” to find the NUMERICAL DERIVATIVE lim f(a + h) – f(a – h) h->0 2h nDeriv f(a) = f(a ) – f(a – 0.001) 0.002

Example 2 Computing a Numerical Derivative Compute nDeriv(x 3, x, 2), the numerical derivative of x 3 at x = 2. (use calculator) Now compute the derivative of f(x) at x = 0 when f(x) = 4x – x 2. (use formula & calculator) How do you know the derivative exists at x = 0?

Homework Page 114 Quick Review 1-10 Exercises 1-19 odds

Warm Up Use the Symmetrical Difference Quotient nDeriv f(a) = f(a ) – f(a – 0.001) To find the numerical derivative of f(x) = 4x – x 2 at x = 1. Check your answer with nderiv(4x – x 2, x, 1)

Example 3 Fooling the Symmetric Difference Quotient Let’s compute NDER(|x|, x, 0) with calculator Using Formula with x = 0 Simplify Check graph, is this answer reasonable?

Graphing a Derivative using nDeriv Let f(x) = ln x Graph f’(x) y = nDeriv (ln x, x, x) Can anyone make a conjecture as to the formula for this derivative function? You try # 27

Theorem 1 Differentiability Implies Continuity If f has a derivative at x = a, then f is continuous at x = a. Is the converse also true? If is continuous at x = a, then f has a derivative at x = a

Theorem 2 Intermediate Value Theorem for Derivatives If a and b are any two points on an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b). Example 5 When considering Theorem 2, does any function have the unit step function as its derivative? U(x) = -1, x < 0 Graph looks like: 1, x ≥ 0

The question of when a function is a derivative of some function is one of the central questions of calculus. We will see the answer in chapter 5! Summarize When will f’(a) fail to exist? Does the calculator always yield reliable results? Explain Describe the Intermediate Value Theorem for derivatives.

Today’s Agenda Warm Up: Page 115 Exercises Correct Homework: Q & A Teamwork: page 115 Exploration #47 Homework: page 114 Exercises 21-35