Let
be a function and
be in its domain. If
is differentiable at
then
is continuous at
Proof
If
is differentiable at
then
exists and
We want to show that
is continuous at
by showing that
Thus,
Therefore, since
is defined and
we conclude that
is continuous at
□
We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function
This function is continuous everywhere; however,
is undefined. This observation leads us to believe that continuity does not imply differentiability. Let’s explore further. For
Let’s consider some additional situations in which a continuous function fails to be differentiable. Consider the function
Thus
does not exist. A quick look at the graph of
clarifies the situation. The function has a vertical tangent line at
(
[link] ).
The function
also has a derivative that exhibits interesting behavior at
We see that
This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero (
[link] ).
In summary:
We observe that if a function is not continuous, it cannot be differentiable, since every differentiable function must be continuous. However, if a function is continuous, it may still fail to be differentiable.
We saw that
failed to be differentiable at
because the limit of the slopes of the tangent lines on the left and right were not the same. Visually, this resulted in a sharp corner on the graph of the function at
From this we conclude that in order to be differentiable at a point, a function must be “smooth” at that point.
As we saw in the example of
a function fails to be differentiable at a point where there is a vertical tangent line.
As we saw with
a function may fail to be differentiable at a point in more complicated ways as well.
A piecewise function that is continuous and differentiable
A toy company wants to design a track for a toy car that starts out along a parabolic curve and then converts to a straight line (
[link] ). The function that describes the track is to have the form
where
and
are in inches. For the car to move smoothly along the track, the function
must be both continuous and differentiable at
Find values of
and
that make
both continuous and differentiable.
For the function to be continuous at
Thus, since
and
we must have
Equivalently, we have
For the function to be differentiable at
must exist. Since
is defined using different rules on the right and the left, we must evaluate this limit from the right and the left and then set them equal to each other:
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
and last as a parent. I know sometimes it's overwhelming to guide a special kid.
but trust the process and be strong and patient as a parent.
Jharna
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