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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.R.118b
Chapter 4, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Properties of Functions

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.
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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.
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Related Practice
Textbook Question

{Use of Tech } Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)

Textbook Question

{Use of Tech} Newton’s method Use Newton’s method to approximate the roots of ƒ(x) = e⁻²ˣ + 2eˣ - 6 to six digits.

Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 


lim_x→0 csc x sin⁻¹ x

Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = x³ ln x on (0, ∞)

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Textbook Question

Maximum printable area A rectangular page in a text (with width x and length y) has an area of 98 in² , top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?

Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

c. Give the approximate coordinates of the inflection point(s) of f.