Domanda del libro di testo
Exponential growth rates
a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 121b
Exponential growth rates
b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.
Guida verificata passo dopo passo1
To compare the growth rates of \( e^x \) and \( e^{ax} \) as \( x \to \infty \), we first need to understand the behavior of exponential functions. Both functions are exponential, but they have different exponents.
Consider the function \( e^x \). As \( x \to \infty \), \( e^x \) grows exponentially without bound. The base \( e \) is a constant greater than 1, which means the function increases rapidly.
Now, consider the function \( e^{ax} \). Here, \( a \) is a positive constant greater than 0. The exponent \( ax \) means that the rate of growth is scaled by \( a \). If \( a > 1 \), \( e^{ax} \) grows faster than \( e^x \). If \( 0 < a < 1 \), \( e^{ax} \) grows slower than \( e^x \).
To compare the growth rates more formally, we can take the limit of the ratio of the two functions as \( x \to \infty \): \( \lim_{x \to \infty} \frac{e^{ax}}{e^x} = \lim_{x \to \infty} e^{(a-1)x} \).
Evaluate the limit: If \( a > 1 \), \( e^{(a-1)x} \to \infty \), indicating \( e^{ax} \) grows faster. If \( 0 < a < 1 \), \( e^{(a-1)x} \to 0 \), indicating \( e^x \) grows faster. If \( a = 1 \), the limit is 1, indicating both grow at the same rate.
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Exponential Functions
Exponential functions are mathematical expressions of the form f(x) = a * e^(bx), where 'e' is the base of the natural logarithm, approximately equal to 2.71828. These functions exhibit rapid growth or decay, depending on the sign of 'b'. Understanding their behavior as x approaches infinity is crucial for analyzing growth rates.
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Exponential Functions
Growth Rate Comparison
When comparing growth rates of functions, we often look at their limits as x approaches infinity. For the functions e^x and e^(ax) where a > 0, we can determine which function grows faster by evaluating the limit of their ratio. This comparison reveals that e^(ax) grows faster than e^x as x becomes very large.
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Percorso guidato
04:16Intro To Related Rates
Limit Analysis
Limit analysis is a fundamental concept in calculus that helps us understand the behavior of functions as they approach a certain point, often infinity. By applying limit techniques, we can rigorously determine the growth rates of functions like e^x and e^(ax) as x approaches infinity, providing insights into their relative rates of increase.
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Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .
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Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?
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