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Arrange Calculus, Algebra, and Special Values from Least to Greatest
Mathematics
Grade 11 of Senior High School
Question Content
Arrange the following values from least to greatest: $\sqrt{3}$, $2!$, $\int_{1}^{5} xdx$, $\sum_{i=3}^{8} i$, $\frac{6\pi}{5}$, $\frac{2}{3}$, $\infty$, $\log_{3}(20)$, $e^4$
Correct Answer
$\\frac{2}{3} < \\sqrt{3} < 2! < \\log_{3}(20) < \\frac{6\\pi}{5} < \\int_{1}^{5} xdx < \\sum_{i=3}^{8} i < e^4 < \\infty$
Detailed Solution Steps
1
Step 1: Calculate the numerical value of each expression:
2
- $\\frac{2}{3} \\approx 0.667$
3
- $\\sqrt{3} \\approx 1.732$
4
- $2! = 2 \\times 1 = 2$
5
- $\\log_{3}(20)$: Since $3^2=9$, $3^3=27$, so $\\log_{3}(20) \\approx 2.727$
6
- $\\frac{6\\pi}{5} \\approx \\frac{6 \\times 3.1416}{5} \\approx 3.770$
7
- $\\int_{1}^{5} xdx$: Use the power rule of integration $\\int x^n dx = \\frac{x^{n+1}}{n+1}+C$, so $\\int_{1}^{5} xdx = \\frac{1}{2}x^2\\big|_{1}^{5} = \\frac{1}{2}(25-1) = 12$
8
- $\\sum_{i=3}^{8} i = 3+4+5+6+7+8 = 33$
9
- $e^4 \\approx 2.718^4 \\approx 54.598$
10
- $\\infty$ represents an infinitely large value, larger than all finite numbers
11
Step 2: Compare the calculated values and arrange them from smallest to largest
Knowledge Points Involved
1
Factorial Calculation
The factorial of a positive integer $n$ (denoted $n!$) is the product of all positive integers from 1 to $n$, defined as $n! = n \\times (n-1) \\times ... \\times 1$, with $0! = 1! = 1$. It is used in combinatorics, probability, and series expansions.
2
Definite Integration (Power Rule)
The power rule for integration states $\\int x^n dx = \\frac{x^{n+1}}{n+1} + C$ for $n \\neq -1$. For definite integrals $\\int_{a}^{b} x^n dx = \\frac{x^{n+1}}{n+1}\\big|_{a}^{b} = \\frac{b^{n+1}-a^{n+1}}{n+1}$, used to calculate the area under a polynomial curve between two bounds.
3
Finite Summation
A finite summation $\\sum_{i=a}^{b} i$ is the sum of consecutive integers from $a$ to $b$. It can be calculated directly by adding terms, or using the formula for the sum of the first $n$ integers: $\\sum_{i=1}^{n}i = \\frac{n(n+1)}{2}$, then adjusting for the starting index.
4
Logarithm and Exponential Values
Logarithms $\\log_b(a)$ represent the exponent to which $b$ must be raised to get $a$. Natural exponential $e^x$ uses Euler's number $e \\approx 2.718$, and grows exponentially, making $e^x$ very large for positive $x>3$. We estimate these values by comparing to known powers of the base.
5
Infinity Concept
$\\infty$ (infinity) is not a real number, but a concept representing a value that is larger than any finite real number. It is always greater than all finite numerical values in ordering problems.
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