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
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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.
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.
Consider the function
U(x) = -1, x < 0 Graph
1, x ≥ 0
Find the right and left hand limits.
Find all points in the domain of f(x)=|x - 2| + 3 where f is not differentiable.
Think graphically first.
You try exercise 1
Differentiable functions include
*Our job this chapter is to learn to find derivatives of all these types of functions!
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!
To evaluate at a point x = a
lim f(a + h) – f(a – h)
h->0 2h
nDeriv f(a) = f(a + 0.001) – f(a – 0.001)
0.002
x = 0?
Page 114
Quick Review 1-10
Exercises 1-19 odds
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)
y = nDeriv (ln x, x, x)
Can anyone make a conjecture as to the formula for this derivative function?
You try # 27
If is continuous at x = a,
then f has a derivative at x = a
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.
Warm Up: Page 115 Exercises 40-43
Correct Homework: Q & A
Teamwork: page 115 Exploration #47
Homework: page 114 Exercises 21-35