Transcript Slide 1

Chapter 3
Derivatives
Section 3.2
Differentiability
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Quick Review
In Exercises 1 – 5, tell whether the limit could be used to define f   a 
 assuming that f is differentiable at a  .
f a  h  f a
1.
lim
3.
lim
5.
lim
h 0
xa
h 0
h
f  x  f a
xa
f a  h  f a  h
2.
lim
4.
lim
h 0
x a
f a  h  f h
h
f a  f  x
ax
h
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Slide 3.2- 2
Quick Review
6.
4
3
Find the domain of the function y  x .
3
4
7.
Find the domain of the function y  x .
8.
Find the range of the function y  x  2  3.
9.
Find the slope of the line y  5  3.2  x    .
10. If f  x   5 x, find
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f  3  0.001  f  3  0.001
0.002
Slide 3.2- 3
Quick Review Solutions
In Exercises 1 – 5, tell whether the limit could be used to define f   a 
 assuming that f is differentiable at a  .
f a  h  f a
1.
lim
3.
lim
5.
lim
h 0
x a
h 0
h
f  x  f a
Yes
2.
lim
Yes
4.
lim
xa
f a  h  f a  h
h
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h 0
x a
f a  h  f h
h
f a  f  x
ax
No
Yes
No
Slide 3.2- 4
Quick Review Solutions
6.
4
3
Find the domain of the function y  x . All Reals
3
4
7.
Find the domain of the function y  x . [0, )
8.
Find the range of the function y  x  2  3. [3, )
9.
Find the slope of the line y  5  3.2  x    . 3.2
10. If f  x   5 x, find
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f  3  0.001  f  3  0.001
0.002
5
Slide 3.2- 5
What you’ll learn about





How f ′(a) Might Fail to Exist
Differentiability Implies Local Linearity
Derivatives on a Calculator
Differentiability Implies Continuity
Intermediate Value Theorem for Derivatives
… and why
Graphs of differentiable functions can be approximated
by their tangent lines at points where the derivative
exists.
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Slide 3.2- 6
How f′(a) Might Fail to Exist
A function will not have a derivative at a point P  a, f  a  
where the slopes of the secant lines,
f  x  f a
xa
fail to approach a limit as x approaches a.
The next figures illustrate four different instances where this occurs.
For example, a function whose graph is otherwise smooth will fail to
have a derivative at a point where the graph has:
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Slide 3.2- 7
How f′(a) Might Fail to Exist
1. a corner, where the one-sided derivatives differ;
f  x  x
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Slide 3.2- 8
How f′(a) Might Fail to Exist
2. a cusp, where the slopes of the secant lines approach  from one side and
approach   from the other (an extreme case of a corner);
f  x  x
2
3
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Slide 3.2- 9
How f′(a) Might Fail to Exist
3. A vertical tangent, where the slopes of the secant lines approach
either  or   from both sides;
f  x  3 x
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Slide 3.2- 10
How f′(a) Might Fail to Exist
4. a discontinuity (which will cause one or both of the one-sided
derivatives to be nonexistent).
1, x  0
U  x  
1, x  0
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Slide 3.2- 11
Example How f′(a) Might Fail to Exist
Show that the function is not differentiable at x  0.
 x3 , x  0
f  x  
4 x, x  0
The right-hand derivative is 4.
The left-hand derivative is 0.
The function is not differentiable at x  0.
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Slide 3.2- 12
How f′(a) Might Fail to Exist
Most of the functions we encounter in calculus are differentiable
wherever they are defined, which means they will not have
corners, cusps, vertical tangent lines or points of discontinuity
within their domains. Their graphs will be unbroken and smooth,
with a well-defined slope at each point.
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Slide 3.2- 13
Differentiability Implies Local Linearity
A good way to think of differentiable functions is that they are
locally linear; that is, a function that is differentiable at a closely
resembles its own tangent line very close to a.
In the jargon of graphing calculators, differentiable curves will
“straighten out” when we zoom in on them at a point of
differentiability.
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Slide 3.2- 14
Differentiability Implies Local Linearity
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Slide 3.2- 15
Derivatives on a Calculator
Many graphing utilities can approximate derivatives numerically with good
accuracy at most points of their domains. For small values of h,
the difference quotient
f a  h  f a
h
is often a good numerical approximation of f   a  .
However, the same value of h will usually yield a better approximation if
we use the symmetric difference quotient
f a  h  f a  h
2h
which is what our graphing calculator uses to calculate NDER f  a  , the
numerical derivative of f at a point a.
The numerical derivative of f as a function is denoted by NDER f  x  .
The numerical derivatives we compute in this book will use h  0.001.
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Slide 3.2- 16
Example Derivatives on a Calculator
Find the numerical derivative of the function f  x   x 2  3
at the point x  2. Use a calculator with h  0.001.
Using a TI-83 Plus we get
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Slide 3.2- 17
Derivatives on a Calculator
Because of the method used internally by the calculator, you will
sometimes get a derivative value at a nondifferentiable point.
This is a case of where you must be “smarter” than the calculator.
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Slide 3.2- 18
Differentiability Implies Continuity
If f has a derivative at x  a, then f is continuous at x  a.
The converse of Theorem 1 is false. A continuous functions
might have a corner, a cusp or a vertical tangent line, and
hence not be differentiable at a given point.
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Slide 3.2- 19
Intermediate Value Theorem for
Derivatives
Not every function can be a derivative.
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  .
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