Domanda del libro di testo
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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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 7.1.72b
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Guida verificata passo dopo passo1
Start with the given function: \(y = 2 - x - x^{3}\), where \(-2 \leq x \leq 2\).
To find the inverse function \(g\), solve the equation \(y = 2 - x - x^{3}\) for \(x\) in terms of \(y\).
Rewrite the equation as \(x^{3} + x = 2 - y\) to isolate terms involving \(x\) on one side.
Recognize that this is a cubic equation in \(x\): \(x^{3} + x - (2 - y) = 0\).
Use an appropriate method (such as the cubic formula or a computer algebra system) to solve for \(x\) as a function of \(y\), and define this solution as the inverse function \(g(y)\).
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Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse g(y) satisfies g(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y, which may require algebraic manipulation and checking domain restrictions to ensure the inverse is well-defined.
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4:49
Inverse Cosine
Derivative of Inverse Functions
The derivative of an inverse function at a point relates to the derivative of the original function by the formula g'(y) = 1 / f'(x), where y = f(x). This relationship helps find the slope of the tangent line to the inverse function without explicitly differentiating it, provided f'(x) ≠ 0 at the point of interest.
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06:35Derivatives of Other Inverse Trigonometric Functions
Tangent Line Approximation
Tangent line approximation uses the derivative at a point to approximate the function near that point with a linear function. For a function f at x₀, the tangent line is y = f(x₀) + f'(x₀)(x - x₀). This concept extends to inverse functions, allowing approximation of g(y) near y₀ using the slope of the inverse function.
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Percorso guidato
05:13Slopes of Tangent Lines
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)
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Domanda del libro di testo
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)
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Domanda del libro di testo
Verify the integration formulas in Exercises 37–40.
37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C
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Domanda del libro di testo
3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)
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Domanda del libro di testo
131. Let f(x) = x * e^(−x).
b. Find all inflection points for f.
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