{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a \u003e 0.b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).

Ch. 4 - Applications of the Derivative

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

문제 4.R.118b

4장, 문제 4.R.118b

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).

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First, identify the domain of the function ƒ(x) = (a + x)ˣ. Since the base (a + x) must be positive, the domain is x > -a.

Consider the behavior of ƒ(x) as x approaches the left boundary of its domain, which is x approaching -a from the right. As x gets closer to -a, the base (a + x) approaches zero, and the exponent x approaches -a. This results in the function approaching zero.

Next, analyze the behavior of ƒ(x) as x approaches infinity. As x becomes very large, both the base (a + x) and the exponent x increase. The function (a + x)ˣ grows rapidly, leading to superexponential growth.

Understand that superexponential growth means the function increases faster than exponential functions, due to the variable being both in the base and the exponent.

Summarize the end behavior: Near the left boundary of its domain, ƒ(x) approaches zero as x approaches -a. As x approaches infinity, ƒ(x) exhibits superexponential growth, increasing very rapidly.

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주요 개념

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Superexponential Functions

Superexponential functions, like f(x) = (a + x)ˣ, grow faster than exponential functions as x increases. This means that as x approaches infinity, the function's value increases at an accelerating rate, which is crucial for understanding its end behavior.

End Behavior

End behavior refers to the behavior of a function as the input approaches certain limits, such as negative or positive infinity. For f(x) = (a + x)ˣ, analyzing end behavior involves determining how the function behaves as x approaches the left boundary of its domain and as x approaches infinity.

Domain of the Function

The domain of a function is the set of all possible input values (x) for which the function is defined. For f(x) = (a + x)ˣ, the domain is all real numbers greater than -a, which affects how we analyze the function's behavior at the left boundary and as x increases.

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).