Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.R.118a
{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.
a. What is the domain of f (in terms of a)?
Verified step by step guidance1
To determine the domain of the function \( f(x) = (a + x)^x \), we need to identify the values of \( x \) for which the expression is defined.
The expression \( (a + x)^x \) is defined when \( a + x > 0 \) because the base of the power must be positive for real number exponents.
Solve the inequality \( a + x > 0 \) to find the values of \( x \). This gives \( x > -a \).
Therefore, the domain of \( f(x) \) in terms of \( a \) is all real numbers \( x \) such that \( x > -a \).
In interval notation, the domain is \( (-a, \infty) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function f(x) = (a + x)ˣ, where a > 0, the domain is determined by the values of x that keep the expression a + x positive, as the base of an exponent must be positive for real-valued outputs.
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Finding the Domain and Range of a Graph
Exponential Functions
Exponential functions are mathematical expressions in the form f(x) = bˣ, where b is a positive constant. In the case of f(x) = (a + x)ˣ, the function exhibits superexponential growth, meaning it grows faster than any polynomial function as x increases. Understanding the behavior of exponential functions is crucial for analyzing their domains and ranges.
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Inequalities
Inequalities are mathematical statements that compare two expressions, indicating whether one is greater than, less than, or equal to the other. In determining the domain of f(x) = (a + x)ˣ, we need to solve the inequality a + x > 0, which leads to x > -a. This understanding of inequalities is essential for identifying valid input values for the function.
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Related Practice
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).
g(x) = x sin⁻¹ x on [-1, 1]
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Textbook Question
{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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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→0 csc x sin⁻¹ x
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Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ dx / (1 - sin² x)
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Textbook Question
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
eˣ and 3ˣ
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