Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
43. f(x) = 5 − 4x, a = 1/2
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.8.2.a
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. 10x^4 + 30x + 1
Verified step by step guidance1
Recall that the function \(e^x\) is an exponential function, which generally grows faster than any polynomial function as \(x \to \infty\).
Identify the given function: \(10x^4 + 30x + 1\), which is a polynomial of degree 4.
Compare the growth rates by considering the limit \(\lim_{x \to \infty} \frac{10x^4 + 30x + 1}{e^x}\).
Since \(e^x\) grows faster than any polynomial, this limit approaches 0, indicating that \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x \to \infty\).
Therefore, the function \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x\) approaches infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Functions
Growth rates describe how functions behave as the input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace as others, especially as x approaches infinity.
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04:16
Intro To Related Rates
Exponential Functions
Exponential functions like e^x grow by continuously multiplying by a constant base raised to the variable power. They increase faster than any polynomial function as x approaches infinity, making them a benchmark for comparing growth rates.
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6:13Exponential Functions
Polynomial Functions
Polynomial functions are sums of terms with variables raised to whole number powers, such as 10x^4 + 30x + 1. Their growth rate is dominated by the highest power term and is slower than exponential functions as x approaches infinity.
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07:00Taylor Polynomials
Related Practice
Textbook Question
5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_3(x)
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. a. arccos(1/2)
Textbook Question
Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?
Textbook Question
a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.
Textbook Question
77. Which one is correct, and which one is wrong? Give reasons for your answers.
a. lim (x → 3) (x - 3) / (x² - 3) = lim (x → 3) 1 / (2x) = 1/6