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Ch. 4 - Applications of the Derivative
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. 4 - Applications of the DerivativeProblema 43
Capitolo 4, Problema 43

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
e⁰·⁰⁶

Guida verificata passo dopo passo
1
Identify the function you want to approximate. In this case, the function is f(x) = e^x.
Choose a value of 'a' that is close to the point you want to estimate, which is x = 0.06. A good choice for 'a' is 0, because e^0 = 1 and it simplifies calculations.
Find the derivative of the function f(x) = e^x. The derivative, f'(x), is also e^x.
Use the formula for linear approximation: L(x) = f(a) + f'(a)(x - a). Substitute a = 0, f(a) = e^0 = 1, and f'(a) = e^0 = 1 into the formula.
Calculate L(0.06) using the linear approximation formula: L(0.06) = 1 + 1 * (0.06 - 0). This will give you an estimate for e^0.06.

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Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is based on the idea that a function can be closely approximated by a linear function when the input is near a specific value. The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x - a), where 'a' is the point of tangency.
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Linearization

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function at that point. For linear approximations, the derivative is crucial as it determines the slope used in the linear equation.
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Derivatives

Exponential Function

The exponential function, denoted as e^x, is a mathematical function that grows rapidly as x increases. It is defined as the function whose derivative is equal to itself, making it unique in calculus. Understanding the properties of the exponential function is essential for estimating values like e^0.06, especially when using linear approximations around a point like e^0.
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