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) = sin 2x + 3 on [-π , π]
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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.89
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
2ˣ and 4ˣ⸍²
Verified step by step guidance1
Step 1: Understand the problem by identifying the two functions given: \( f(x) = 2^x \) and \( g(x) = 4^{x^{1/2}} \). We need to determine which function grows faster as \( x \to \infty \).
Step 2: Simplify the second function \( g(x) = 4^{x^{1/2}} \). Notice that \( 4^{x^{1/2}} = (2^2)^{x^{1/2}} = 2^{2x^{1/2}} \). This means \( g(x) = 2^{2\sqrt{x}} \).
Step 3: Compare the exponents of the base 2 in both functions. For \( f(x) = 2^x \), the exponent is \( x \). For \( g(x) = 2^{2\sqrt{x}} \), the exponent is \( 2\sqrt{x} \).
Step 4: Analyze the growth of the exponents as \( x \to \infty \). The exponent \( x \) in \( f(x) \) grows linearly, while the exponent \( 2\sqrt{x} \) in \( g(x) \) grows as the square root of \( x \), which is slower than linear growth.
Step 5: Conclude that \( f(x) = 2^x \) grows faster than \( g(x) = 4^{x^{1/2}} \) as \( x \to \infty \), because the linear growth of \( x \) in the exponent of \( f(x) \) outpaces the square root growth of \( 2\sqrt{x} \) in the exponent of \( g(x) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. These functions grow rapidly as 'x' increases, and their growth rate is determined by the base 'b'. In this question, the functions 2^x and 4^x are both exponential, but they have different bases, which affects their growth rates.
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Exponential Functions
Growth Rate Comparison
To compare the growth rates of two functions, we often analyze their limits or use derivatives. For exponential functions, a common method is to express them in terms of the same base. In this case, 4^x can be rewritten as (2^2)^x = 2^(2x), allowing for a direct comparison with 2^x. This helps in determining which function grows faster as 'x' approaches infinity.
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Limit Analysis
Limit analysis involves evaluating the behavior of functions as they approach a certain point, often infinity. By calculating the limit of the ratio of two functions, we can determine their relative growth rates. If the limit approaches zero, one function grows slower; if it approaches infinity, the other grows faster; and if it approaches a non-zero constant, they grow at comparable rates.
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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).
ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]
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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 - 6x² on [-1, 5]
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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ ln ((x +1) / (x-1))
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Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ
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Textbook Question
Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>
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