Textbook Question
88. Incorrect Calculation
b. Evaluate ∫(from -1 to 1) dx/x or show that the integral does not exist.
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.7.86b
Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.
b. Is L concave up or concave down on [0, ∞)?
Verified step by step guidance1
Step 1: Recall the formula for the arc length of a curve y = f(x) from x = a to x = b: \( L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx \). Here, \( f(x) = x^2 \), so \( \frac{dy}{dx} = 2x \).
Step 2: Substitute \( \frac{dy}{dx} = 2x \) into the arc length formula. This gives \( L(c) = \int_0^c \sqrt{1 + (2x)^2} dx \), which simplifies to \( L(c) = \int_0^c \sqrt{1 + 4x^2} dx \).
Step 3: To determine whether \( L(c) \) is concave up or concave down, compute the first derivative \( L'(c) \) and the second derivative \( L''(c) \). Start by differentiating \( L(c) \) with respect to \( c \). Using the Fundamental Theorem of Calculus, \( L'(c) = \sqrt{1 + 4c^2} \).
Step 4: Differentiate \( L'(c) = \sqrt{1 + 4c^2} \) to find \( L''(c) \). Use the chain rule: \( L''(c) = \frac{d}{dc} \sqrt{1 + 4c^2} = \frac{1}{2\sqrt{1 + 4c^2}} \cdot 8c \), which simplifies to \( L''(c) = \frac{4c}{\sqrt{1 + 4c^2}} \).
Step 5: Analyze the sign of \( L''(c) \) on \([0, \infty)\). Since \( c \geq 0 \) and \( \sqrt{1 + 4c^2} > 0 \), \( L''(c) > 0 \) for all \( c \geq 0 \). Therefore, \( L(c) \) is concave up on \([0, \infty)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length of a Curve
The arc length of a curve defined by a function f(x) from x = a to x = b is calculated using the formula L = ∫[a to b] √(1 + (f'(x))²) dx. For the parabola f(x) = x², we first need to find its derivative f'(x) = 2x, which will be used in the arc length formula to determine L(c) from 0 to c.
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Arc Length of Parametric Curves
Concavity
Concavity refers to the direction in which a curve bends. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing. Conversely, it is concave down if the second derivative is negative. Understanding concavity helps in analyzing the behavior of the length function L(c) as c varies.
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05:59Determining Concavity Given a Function
Second Derivative Test
The second derivative test is a method used to determine the concavity of a function. By computing the second derivative of L(c), we can assess whether it is positive or negative over the interval [0, ∞). This test is crucial for answering whether L(c) is concave up or down, providing insights into the nature of the length of the parabola as c increases.
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06:02The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question
58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:
b. Use substitution.
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Textbook Question
The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.
The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]
(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project ""The exponential Eiffel Tower"")
b. With a = 0 and c = 2, find the equations of the lines tangent to both curves at x = 0
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Textbook Question
Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].
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Textbook Question
59. Two Methods
b. Evaluate ∫(x / √(x + 1)) dx using substitution.
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
71. Let f(x) = √(sin x).
b. Find an upper bound on the absolute error in the estimate from part (a) using Theorem 8.1. (Hint: |f''''(x)| ≤ 1 on [1,2].)
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