Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problema 101

Capitolo 4, Problema 101

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x²⁰ ; 1.0001ˣ

Guida verificata passo dopo passo

Step 1: To compare the growth rates of the functions \( f(x) = x^{20} \) and \( g(x) = 1.0001^x \), we can use the concept of limits. Specifically, we will evaluate the limit of the ratio \( \frac{f(x)}{g(x)} \) as \( x \to \infty \).

Step 2: Set up the limit expression: \( \lim_{x \to \infty} \frac{x^{20}}{1.0001^x} \). This will help us determine which function grows faster.

Step 3: Analyze the behavior of the numerator \( x^{20} \) and the denominator \( 1.0001^x \) as \( x \to \infty \). The polynomial \( x^{20} \) grows at a polynomial rate, while the exponential function \( 1.0001^x \) grows at an exponential rate.

Step 4: Recall that exponential functions generally grow faster than polynomial functions as \( x \to \infty \). Therefore, we expect \( 1.0001^x \) to outpace \( x^{20} \) in growth.

Step 5: Conclude that if the limit \( \lim_{x \to \infty} \frac{x^{20}}{1.0001^x} = 0 \), then \( g(x) = 1.0001^x \) grows faster than \( f(x) = x^{20} \). If the limit were a non-zero constant, they would have comparable growth rates.

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are crucial for analyzing the behavior of functions at infinity or near specific points, allowing us to determine growth rates and continuity. In this context, limits help compare the growth of the two functions as x approaches infinity.

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Growth Rates

Growth rates describe how quickly a function increases as its input increases. In calculus, we often compare polynomial functions, like x²⁰, with exponential functions, like 1.0001ˣ, to determine which grows faster. Understanding the nature of these functions is essential for evaluating their limits and establishing their relative growth.

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Asymptotic Behavior

Asymptotic behavior refers to the behavior of functions as the input approaches infinity. It helps in classifying functions based on their growth rates, indicating whether one function dominates another in terms of growth. Analyzing the asymptotic behavior of x²⁰ and 1.0001ˣ will reveal which function grows faster as x becomes very large.

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