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Chapter 8: Problem 1

If the interval [4,18] is partitioned into \(n=28\) sub-intervals of equal length, what is \(\Delta x ?\)

Short Answer

Expert verified
Answer: The length of each sub-interval is 0.5.

Step by step solution

01

Determine the length of the entire interval

To find the length of the entire interval [4, 18], we need to subtract the lower endpoint (4) from the upper endpoint (18). So, the length of the interval is: \[L = 18 - 4\]
02

Calculate the length of the interval

Now we can calculate the length of the interval L: \[L = 18 - 4 = 14\]
03

Divide the total length by the number of sub-intervals

To find the length of each sub-interval, or \(\Delta x\), we divide the total length of the interval (14) by the number of sub-intervals (28): \[\Delta x = \frac{L}{n} = \frac{14}{28}\]
04

Calculate the length of each sub-interval

Lastly, we can compute the length of each sub-interval, \(\Delta x\): \[\Delta x = \frac{14}{28} = 0.5\] The length of each sub-interval, \(\Delta x\), is 0.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a mathematical method used to represent a set of numbers that form an interval. It provides a compact and visual way of denoting which numbers are included in a range. There are several types of intervals:
  • Closed Interval [a, b]: Both ends are included. For example, [4, 18] includes all numbers from 4 to 18, including 4 and 18 themselves.
  • Open Interval (a, b): Both ends are excluded. For example, (4, 18) would include numbers greater than 4 and less than 18.
  • Half-Open Intervals [a, b) or (a, b]: One end is included and the other is excluded. [a, b) means including 'a' and excluding 'b', while (a, b] is the opposite.
Understanding interval notation is important for determining the range of numbers included in a calculation or equation. It's a staple in solving mathematical problems that involve continuous ranges of numbers.
Easy to read and use, this notation helps us quickly see which numbers are relevant for our calculations and analyses.
Sub-Intervals
Sub-intervals are smaller or shorter intervals into which a main interval is divided. They are particularly useful in numerical methods, data partitioning, and statistical calculations.
When you divide an interval into sub-intervals, each of these smaller divisions contributes to comprehensive analyses or problem-solving steps. For example, if we partition the interval [4, 18] into 28 equal parts or sub-intervals, each sub-interval represents a small range within the larger interval.
Using sub-intervals can help simplify:
  • Complex Calculations: Breaking down a large range into smaller parts can make calculations more manageable.
  • Data Analysis: In statistics, breaking data into sub-intervals can help in interpreting trends and making predictions.
  • Numerical Accuracy: In simulations, sub-intervals allow us to approximate values more accurately over a continuous range.
By understanding sub-intervals, you divide a problem into smaller, easier to handle parts that can be tackled individually.
Calculating Delta x
Calculating \(\Delta x\) is a fundamental aspect of partitioning intervals, where it represents the length of each sub-interval. This is key in understanding how intervals are broken down into equal parts for various calculations.
To find \(\Delta x\), you follow these simple steps:
  • Determine the Interval Length: Subtract the lower bound from the upper bound. For example, in the interval [4, 18], the length is \(18 - 4 = 14\).
  • Divide the Length by Number of Sub-Intervals: Use the formula \(\Delta x = \frac{L}{n}\), where \(L\) is the interval length and \(n\) is the number of sub-intervals. Here, you divide 14 by 28, giving you \(\Delta x = \frac{14}{28} = 0.5\).
This calculation ensures that each sub-interval is evenly spaced, which is essential in analyses that require uniformity, such as in integration and numerical methods.
By understanding \(\Delta x\), we can accurately and effectively partition intervals for various mathematical and real-world applications.

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Most popular questions from this chapter

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution \(u=\tan (x / 2)\) or, equivalently, \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ $$\text { Evaluate } \int_{0}^{\pi / 3} \frac{\sin \theta}{1-\sin \theta} d \theta$$.

Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=e^{x^{2}}\) a. Find a Trapezoid Rule approximation to \(\int_{0}^{1} e^{x^{2}} d x\) using \(n=50\) subintervals. b. Calculate \(f^{-}(x)\) c. Explain why \(\left|f^{*}(x)\right|<18\) on [0,1] , given that \(e<3\). d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77 ). a. \(\int x^{4} e^{x} d x \quad\) b. \(\int 7 x e^{3 x} d x\) c. \(\int_{-1}^{0} 2 x^{2} \sqrt{x+1} d x\) d. \(\int\left(x^{3}-2 x\right) \sin 2 x \, d x\) e. \(\int \frac{2 x^{2}-3 x}{(x-1)^{3}} d x\) f. \(\int \frac{x^{2}+3 x+4}{\sqrt[3]{2 x+1}} d x\) g. Why doesn't tabular integration work well when applied to \(\int \frac{x}{\sqrt{1-x^{2}}} d x \, ?\) Evaluate this integral using a different method.

Let \(a>0\) and let \(R\) be the region bounded by the graph of \(y=e^{-a x}\) and the \(x\) -axis on the interval \([b, \infty)\). a. Find \(A(a, b),\) the area of \(R\) as a function of \(a\) and \(b\). b. Find the relationship \(b=g(a)\) such that \(A(a, b)=2\). c. What is the minimum value of \(b\) (call it \(b^{*}\) ) such that when \(b>b^{*}, A(a, b)=2\) for some value of \(a>0 ?\)

Period of a pendulum A standard pendulum of length \(L\) that swings under the influence of gravity alone (no resistance) has a period of $$ T=\frac{4}{\omega} \int_{0}^{\pi / 2} \frac{d \varphi}{\sqrt{1-k^{2} \sin ^{2} \varphi}} $$ where \(\omega^{2}=g / L, k^{2}=\sin ^{2}\left(\theta_{0} / 2\right), g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(\theta_{0}\) is the initial angle from which the pendulum is released (in radians). Use numerical integration to approximate the period of a pendulum with \(L=1 \mathrm{m}\) that is released from an angle of \(\theta_{0}=\pi / 4\) rad.
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