The properties associated with the summation process are given in the following rule.
Rule: properties of sigma notation
Let
and
represent two sequences of terms and let
c be a constant. The following properties hold for all positive integers
n and for integers
m , with
Proof
We prove properties 2. and 3. here, and leave proof of the other properties to the Exercises.
2. We have
3. We have
□
A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for
sums and powers of integers , and we use them in the next set of examples.
Rule: sums and powers of integers
The sum of
n integers is given by
The sum of consecutive integers squared is given by
The sum of consecutive integers cubed is given by
Evaluation using sigma notation
Write using sigma notation and evaluate:
The sum of the terms
for
The sum of the terms
for
Multiplying out
we can break the expression into three terms.
Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.
Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let
be a continuous, nonnegative function defined on the closed interval
We want to approximate the area
A bounded by
above, the
x -axis below, the line
on the left, and the line
on the right (
[link] ).
How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval
into
n subintervals of equal width,
We do this by selecting equally spaced points
with
and
for
We denote the width of each subinterval with the notation Δ
x , so
and
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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