Textbook Question
13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.AAE.1
Find the limits in Exercises 1–6.
1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)
Verified step by step guidance1
Recognize that the integral \( \int_0^b \frac{dx}{\sqrt{1 - x^2}} \) represents the inverse sine function, since \( \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \).
Rewrite the integral using the antiderivative: \( \int_0^b \frac{dx}{\sqrt{1 - x^2}} = \arcsin b - \arcsin 0 \).
Evaluate \( \arcsin 0 \), which is 0, so the integral simplifies to \( \arcsin b \).
Set up the limit expression as \( \lim_{b \to 1^-} \arcsin b \), where \( b \) approaches 1 from the left side.
Recall the value of \( \arcsin 1 \) and use it to determine the limit of the integral as \( b \) approaches 1 from the left.
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral as a Limit of Riemann Sums
A definite integral represents the accumulation of quantities, such as area under a curve, between two limits. It is defined as the limit of Riemann sums as the partition gets finer. Understanding this helps interpret the integral ∫₀ᵇ 1/√(1-x²) dx as a function of the upper limit b.
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Introduction to Riemann Sums
Limit of a Function from the Left (One-Sided Limit)
A left-hand limit, denoted as lim(b→1⁻), considers values of b approaching 1 from values less than 1. This concept is crucial when the function or integral behaves differently near the boundary, ensuring the limit is evaluated correctly as b approaches 1 from below.
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Integral of 1/√(1 - x²) and its Connection to Inverse Trigonometric Functions
The integral of 1/√(1 - x²) with respect to x is arcsin(x) + C. Recognizing this allows direct evaluation of the definite integral by applying the Fundamental Theorem of Calculus, simplifying the limit problem to evaluating arcsin(b) as b approaches 1 from the left.
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Related Practice
Textbook Question
Find the limits in Exercises 1–6.
3. lim(x→0⁺) (cox(√x))^(1/x)
Textbook Question
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
Textbook Question
Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?
Textbook Question
Find the limits in Exercises 1–6.
5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))
Textbook Question
19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.