One application that helps illustrate the Mean Value Theorem involves velocity. For example, suppose we drive a car for 1 h down a straight road with an average velocity of 45 mph. Let
and
denote the position and velocity of the car, respectively, for
h. Assuming that the position function
is differentiable, we can apply the Mean Value Theorem to conclude that, at some time
the speed of the car was exactly
Mean value theorem and velocity
If a rock is dropped from a height of 100 ft, its position
seconds after it is dropped until it hits the ground is given by the function
Determine how long it takes before the rock hits the ground.
Find the average velocity
of the rock for when the rock is released and the rock hits the ground.
Find the time
guaranteed by the Mean Value Theorem when the instantaneous velocity of the rock is
When the rock hits the ground, its position is
Solving the equation
for
we find that
Since we are only considering
the ball will hit the ground
sec after it is dropped.
The average velocity is given by
The instantaneous velocity is given by the derivative of the position function. Therefore, we need to find a time
such that
Since
is continuous over the interval
and differentiable over the interval
by the Mean Value Theorem, there is guaranteed to be a point
such that
Taking the derivative of the position function
we find that
Therefore, the equation reduces to
Solving this equation for
we have
Therefore,
sec after the rock is dropped, the instantaneous velocity equals the average velocity of the rock during its free fall:
ft/sec.
Suppose a ball is dropped from a height of 200 ft. Its position at time
is
Find the time
when the instantaneous velocity of the ball equals its average velocity.
Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections.
At this point, we know the derivative of any constant function is zero. The Mean Value Theorem allows us to conclude that the converse is also true. In particular, if
for all
in some interval
then
is constant over that interval. This result may seem intuitively obvious, but it has important implications that are not obvious, and we discuss them shortly.
Corollary 1: functions with a derivative of zero
Let
be differentiable over an interval
If
for all
then
constant for all
Proof
Since
is differentiable over
must be continuous over
Suppose
is not constant for all
in
Then there exist
where
and
Choose the notation so that
Therefore,
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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