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).

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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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주요 개념

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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.

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.

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.

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).