Calculus, Part 4
Question 1: Explain the Trapezoidal Rule.
Answer 1: The trapezoidal rule, also called the trapezoid rule or the trapezium rule, is another way to approximate the area under a curve. In this case, a trapezoid is drawn such that one side of the trapezoid is on the x-axis and the parallel sides terminate on points of the curve so that part of the curve is above the trapezoid and part of the curve is below the trapezoid. While this does not give an exact area for the region under the curve, it does provide a close approximation.
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Question 2: Define sequence and element.
Answer 2: A function with a domain comprised of the set of positive integers is a sequence. Each member of the sequence is an element, or individual term. Each element of a sequence is identified by the notation ?, where a is the term of the sequence, and n is the integer identifying which term in the sequence a is. There are two different ways to represent a sequence that contains the element ?. The first is the simple notation {?}. The expanded notation of a sequence is ?1, ?2, ?3, …?, … . Notice that the expanded form does not end with the nth term. There is no indication that the nth term is the last term in the sequence, only that the nth term is an element of the sequence.
Question 3: Discuss the terms limit, converge, and diverge as they relate to sequences.
Answer 3: Some sequences will have a limit, or a value the sequence approaches but never passes. A sequence that has a limit is known as a convergent sequence because all the values of the sequence seemingly converge at that point. Sequences that do not converge at a particular limit are divergent sequences. The easiest way to determine whether a sequence converges or diverges is to find the limit of the sequence. If the limit is a real number, the sequence is a convergent sequence. If the limit is infinity, the sequence is a divergent sequence. Remember the following rules for finding limits:lim?8?=? for all real numbers klim?81?=0 lim?8?=0 for all real numbers k and positive rational numbers p.
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