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 7.8.2.a
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
Guida verificata passo dopo passo1
Recall that the function \(e^x\) is an exponential function, which generally grows faster than any polynomial function as \(x \to \infty\).
Identify the given function: \(10x^4 + 30x + 1\), which is a polynomial of degree 4.
Compare the growth rates by considering the limit \(\lim_{x \to \infty} \frac{10x^4 + 30x + 1}{e^x}\).
Since \(e^x\) grows faster than any polynomial, this limit approaches 0, indicating that \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x \to \infty\).
Therefore, the function \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x\) approaches infinity.
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Growth Rates of Functions
Growth rates describe how functions behave as the input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace as others, especially as x approaches infinity.
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Percorso guidato
04:16
Intro To Related Rates
Exponential Functions
Exponential functions like e^x grow by continuously multiplying by a constant base raised to the variable power. They increase faster than any polynomial function as x approaches infinity, making them a benchmark for comparing growth rates.
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6:13Exponential Functions
Polynomial Functions
Polynomial functions are sums of terms with variables raised to whole number powers, such as 10x^4 + 30x + 1. Their growth rate is dominated by the highest power term and is slower than exponential functions as x approaches infinity.
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07:00Taylor Polynomials
Pratica correlata
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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5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_3(x)
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a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.
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Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?
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