Domanda del libro di testo
In Exercises 41–44:
a. Find f⁻¹(x).
41. f(x) = 2x + 3, a = −1
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.3.145a
145. The linearization of eˣ at x = 0
a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.
Guida verificata passo dopo passo1
Recall that the linearization (or linear approximation) of a function \( f(x) \) at a point \( a \) is given by the formula:
\[ L(x) = f(a) + f'(a)(x - a) \]
This represents the equation of the tangent line to the curve at \( x = a \).
Identify the function and the point of linearization: here, \( f(x) = e^x \) and \( a = 0 \).
Calculate \( f(a) = f(0) = e^0 \). Since \( e^0 = 1 \), this gives the value of the function at the point of linearization.
Find the derivative of the function: \( f'(x) = \frac{d}{dx} e^x = e^x \). Then evaluate it at \( x = 0 \), so \( f'(0) = e^0 = 1 \).
Substitute these values into the linearization formula:
\[ L(x) = f(0) + f'(0)(x - 0) = 1 + 1 \cdot x = 1 + x \]
This is the linear approximation of \( e^x \) near \( x = 0 \).
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Linearization of a Function
Linearization approximates a function near a point using the tangent line at that point. It replaces the function with a linear expression that is easier to work with, typically written as L(x) = f(a) + f'(a)(x - a), where a is the point of approximation.
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Percorso guidato
07:17
Linearization
Derivative of eˣ
The derivative of the exponential function eˣ is eˣ itself. At x = 0, this derivative equals 1, which is essential for finding the slope of the tangent line used in the linear approximation.
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Percorso guidato
05:44Derivatives
Evaluating Functions and Derivatives at a Point
To find the linear approximation at x = 0, you must evaluate both the function and its derivative at that point. For eˣ, e⁰ = 1 and the derivative at 0 is also 1, leading to the linearization formula eˣ ≈ 1 + x.
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Percorso guidato
4:26Evaluating Composed Functions
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
67. ∫(from 0 to 2√3)dx/√(4+x²)
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Domanda del libro di testo
In Exercises 41–44:
a. Find f⁻¹(x).
42. f(x) = (x + 2) / (1 − x), a = 1/2
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Domanda del libro di testo
In Exercises 41–44:
a. Find f⁻¹(x).
43. f(x) = 5 − 4x, a = 1/2
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Domanda del libro di testo
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. 10x^4 + 30x + 1
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Domanda del libro di testo
21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).
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