Textbook Question
76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x
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 8.8.67b
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
b. Calculate f''(x).
Verified step by step guidance1
Step 1: Recall the function f(x) = √(x³ + 1). To calculate f''(x), we first need to find f'(x), the first derivative of f(x). Use the chain rule for differentiation since f(x) involves a composition of functions.
Step 2: Start by differentiating the outer function √(u), where u = x³ + 1. The derivative of √(u) is (1 / (2√(u))) * du/dx. Substitute u = x³ + 1 into this formula.
Step 3: Differentiate the inner function x³ + 1 with respect to x. The derivative of x³ is 3x², and the derivative of 1 is 0. Therefore, du/dx = 3x².
Step 4: Combine the results from Step 2 and Step 3 to find f'(x). Substitute du/dx = 3x² into the formula for the derivative of √(u). This gives f'(x) = (1 / (2√(x³ + 1))) * 3x².
Step 5: To find f''(x), differentiate f'(x) = (3x² / (2√(x³ + 1))) using the quotient rule. The quotient rule states that if h(x) = g(x) / k(x), then h'(x) = (g'(x)k(x) - g(x)k'(x)) / [k(x)]². Apply this rule carefully to f'(x), treating the numerator and denominator separately.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Second Derivative
The second derivative of a function, denoted as f''(x), measures the rate of change of the first derivative, f'(x). It provides information about the concavity of the function: if f''(x) is positive, the function is concave up, and if negative, it is concave down. This concept is crucial for understanding the behavior of functions and for applications in optimization and curve sketching.
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The Second Derivative Test: Finding Local Extrema
Theorem 8.1
Theorem 8.1 typically refers to a specific theorem in calculus that deals with estimating errors in approximations, often related to Taylor series or numerical methods. Understanding this theorem is essential for evaluating how closely a function can be approximated by its derivatives, which is particularly relevant when calculating the second derivative and assessing the accuracy of such calculations.
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06:11Fundamental Theorem of Calculus Part 1
Function Composition and Roots
The function f(x) = √(x³ + 1) involves both composition and roots. Understanding how to differentiate composite functions and apply the chain rule is vital for finding derivatives. Additionally, recognizing how to manipulate roots and powers is necessary for simplifying expressions and performing calculations accurately, especially when deriving higher-order derivatives.
Recommended video:
3:48Evaluate Composite Functions - Special Cases
Related Practice
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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Textbook Question
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, ∞)?
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Textbook Question
63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. If m is a positive integer, then ∫[0 to π] sin^m(x) dx = 0.
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
Textbook Question
82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.
b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.