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### 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) = √(x2 + 0.0001) + 0.99

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 + 0.001) – f(a – 0.001)

0.002

Example 2Computing a Numerical Derivative

- Compute nDeriv(x3, x, 2), the numerical derivative of x3 at x = 2. (use calculator)
- Now compute the derivative of f(x) at x = 0 when f(x) = 4x – x2. (use formula & calculator)
- How do you know the derivative exists at

x = 0?

Warm Up

Use the Symmetrical Difference Quotient

nDeriv f(a) = f(a + 0.001) – f(a – 0.001)

0.002

To find the numerical derivative of

f(x) =4x – x2 at x = 1.

Check your answer with nderiv(4x – x2, x, 1)

Example 3Fooling 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 1Differentiability 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 2Intermediate 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 40-43

Correct Homework: Q & A

Teamwork: page 115 Exploration #47

Homework: page 114 Exercises 21-35

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