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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 59
Capitolo 3, Problema 59

Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.

Guida verificata passo dopo passo
1
Step 1: Understand the problem. We need to find a value of 'a' such that the derivative of the piecewise function f(x) is continuous at x=1. The function is defined as f(x) = 2x^2 for x ≤ 1 and f(x) = ax - 2 for x > 1.
Step 2: Ensure f(x) is continuous at x=1. For f(x) to be continuous at x=1, the left-hand limit (as x approaches 1 from the left) must equal the right-hand limit (as x approaches 1 from the right) and also equal f(1). Calculate these limits and set them equal to each other.
Step 3: Differentiate each piece of the function. Find f'(x) for x ≤ 1 and for x > 1. For x ≤ 1, differentiate 2x^2 to get f'(x) = 4x. For x > 1, differentiate ax - 2 to get f'(x) = a.
Step 4: Ensure f'(x) is continuous at x=1. For f'(x) to be continuous at x=1, the left-hand derivative (as x approaches 1 from the left) must equal the right-hand derivative (as x approaches 1 from the right). Set 4(1) equal to a.
Step 5: Solve for 'a'. From the equation 4 = a, determine the value of 'a' that makes f'(x) continuous at x=1.

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Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, f(x) has two distinct definitions: one for x ≤ 1 and another for x > 1. Understanding how to evaluate and analyze these functions at the boundaries is crucial for determining continuity and differentiability.
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Piecewise Functions

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For f(x) to be continuous at x = 1, the left-hand limit (as x approaches 1 from the left) must equal the right-hand limit (as x approaches 1 from the right) and also equal f(1).
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Intro to Continuity

Differentiability and Its Relationship to Continuity

A function is differentiable at a point if it has a defined derivative there, which requires the function to be continuous at that point. For f' to be continuous at x = 1, not only must f be continuous at that point, but the derivatives from both sides must also match. This involves calculating f'(x) for both pieces and ensuring they are equal at x = 1.
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Intro to Continuity
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