Domanda del libro di testo
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = 1/x², x > 0
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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 7.3.130
130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?
Guida verificata passo dopo passo1
Identify the function given: \(f(x) = 2e^{\sin(\frac{x}{2})}\), which is periodic due to the sine function inside the exponent.
To find the extreme values, compute the derivative \(f'(x)\) using the chain rule: \(f'(x) = 2e^{\sin(\frac{x}{2})} \cdot \cos(\frac{x}{2}) \cdot \frac{1}{2}\).
Set the derivative equal to zero to find critical points: \(f'(x) = 0 \Rightarrow 2e^{\sin(\frac{x}{2})} \cdot \cos(\frac{x}{2}) \cdot \frac{1}{2} = 0\). Since \(2e^{\sin(\frac{x}{2})} \neq 0\), this simplifies to \(\cos(\frac{x}{2}) = 0\).
Solve \(\cos(\frac{x}{2}) = 0\) for \(x\): \(\frac{x}{2} = \frac{\pi}{2} + k\pi\), where \(k\) is any integer, so \(x = \pi + 2k\pi\).
Evaluate \(f(x)\) at these critical points to find the extreme values: \(f(x) = 2e^{\sin(\frac{x}{2})}\), and since \(\sin(\frac{x}{2})\) oscillates between -1 and 1, the minimum and maximum values correspond to \(\sin(\frac{x}{2}) = -1\) and \(\sin(\frac{x}{2}) = 1\), respectively.
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Periodic Functions
A periodic function repeats its values at regular intervals called periods. Understanding the period helps identify where the function's behavior repeats, which is essential for locating extreme values within one period and extending that knowledge to all periods.
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Percorso guidato
6:22
Graphs of Secant and Cosecant Functions
Extreme Values of a Function
Extreme values are points where a function reaches local maxima or minima. These occur where the derivative is zero or undefined, and the function changes direction. Finding these points involves differentiating the function and solving for critical points.
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06:37Average Value of a Function
Chain Rule and Differentiation of Composite Functions
The function f(x) = 2e^(sin(x/2)) is a composite function involving an exponential and a trigonometric function. Differentiating it requires applying the chain rule, which handles the derivative of a function inside another function, crucial for finding critical points.
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03:58The Chain Rule for 3+ Functions
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