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)
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 7.6.5a
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. a. arccos(1/2)
Verified step by step guidance1
Recall that the function arccos(x) gives the angle \( \theta \) whose cosine is \( x \), and its principal value range is \( 0 \leq \theta \leq \pi \) (or 0 to 180 degrees).
Identify the value inside the arccos function: here it is \( \frac{1}{2} \). We want to find \( \theta \) such that \( \cos(\theta) = \frac{1}{2} \).
Use a reference triangle to determine the angle with cosine \( \frac{1}{2} \). Recall that in a right triangle, cosine is adjacent over hypotenuse, so the adjacent side is half the hypotenuse.
From common special angles, recognize that \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). Since arccos returns values in the first and second quadrants, the angle is either \( \frac{\pi}{3} \) or \( \pi - \frac{\pi}{3} = \frac{2\pi}{3} \).
Conclude that the principal value of \( \arccos(\frac{1}{2}) \) is \( \frac{\pi}{3} \), and if considering the full range of cosine, the other angle in the second quadrant is \( \frac{2\pi}{3} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, like arccos, return the angle whose trigonometric ratio matches a given value. For arccos(1/2), it finds the angle whose cosine is 1/2, typically within the principal range of 0 to π radians (0° to 180°). Understanding their ranges is essential to identify correct angle values.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions
Reference Triangles
Reference triangles are right triangles used to relate trigonometric ratios to angles in different quadrants. By using the known side ratios, you can find the acute reference angle and then determine the actual angle based on the quadrant specified, ensuring correct angle measurement.
Recommended video:
6:04Introduction to Trigonometric Functions
Quadrants and Angle Sign Conventions
The coordinate plane is divided into four quadrants, each with specific signs for sine, cosine, and tangent. Knowing which quadrant an angle lies in helps determine the correct angle value from the reference angle, especially since cosine is positive in the first and fourth quadrants.
Recommended video:
5:30Trig Values in Quadrants II, III, & IV
Related Practice
Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))
Textbook Question
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + 4x
Textbook Question
9. True, or false? As x→∞,
a. x = o(x)
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Textbook Question
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
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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