82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

x¹⸍² and x¹⸍³

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Step 1: Understand the concept of growth rates. In calculus, when comparing growth rates of functions, we often look at their behavior as x approaches infinity.

Step 2: Consider the functions given: x^(1/2) and x^(1/3). These are both power functions, and their growth rates can be compared by examining their exponents.

Step 3: Recall that for power functions of the form x^a, the function with the larger exponent a will grow faster as x approaches infinity.

Step 4: Compare the exponents of the two functions: 1/2 and 1/3. Since 1/2 is greater than 1/3, the function x^(1/2) grows faster than x^(1/3) as x approaches infinity.

Step 5: Conclude that x^(1/2) grows faster than x^(1/3) based on the comparison of their exponents.

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Growth Rates of Functions

The growth rate of a function describes how quickly the function's value increases as the input variable grows. In calculus, we often compare growth rates using limits, derivatives, or asymptotic analysis to determine which function increases faster as x approaches infinity.

Limits and Asymptotic Behavior

Limits are fundamental in calculus for analyzing the behavior of functions as they approach a certain point or infinity. Asymptotic behavior refers to the trend of a function as the input becomes very large, allowing us to compare functions by examining their limits at infinity to see which dominates.

Power Functions

Power functions are expressions of the form f(x) = x^n, where n is a real number. The growth rate of power functions is determined by the exponent n; specifically, for large values of x, a function with a higher exponent will grow faster than one with a lower exponent, which is crucial for comparing x^(1/2) and x^(1/3).

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.x¹⸍² and x¹⸍³