Textbook Question
86. This exercise explores the difference between
lim(x→∞)(1 + 1/x²)^x
and
lim(x→∞)(1 + 1/x)^x = e
c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.
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 7.6.7c
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. c. arcsec(-2)
Verified step by step guidance1
Recall that the function \( \arcsec(x) \) is the inverse of the secant function, so \( \arcsec(-2) \) is the angle \( \theta \) such that \( \sec(\theta) = -2 \).
Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), rewrite the equation as \( \frac{1}{\cos(\theta)} = -2 \), which implies \( \cos(\theta) = -\frac{1}{2} \).
Identify the quadrants where \( \cos(\theta) \) is negative. Cosine is negative in Quadrants II and III, so the angle \( \theta \) must lie in one of these quadrants.
Use a reference triangle to find the reference angle \( \alpha \) where \( \cos(\alpha) = \frac{1}{2} \). The reference angle \( \alpha \) corresponds to \( \frac{\pi}{3} \) (or 60 degrees).
Determine the actual angle \( \theta \) in the appropriate quadrant(s) by adjusting the reference angle: in Quadrant II, \( \theta = \pi - \alpha \); in Quadrant III, \( \theta = \pi + \alpha \). Since the principal value of \( \arcsec(x) \) is usually taken in \([0, \pi]\) excluding \( \frac{\pi}{2} \), select the angle in Quadrant II.
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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 arcsec, return the angle whose trigonometric value matches the given input. For arcsec(x), it gives the angle whose secant is x. Understanding their domains and ranges is essential to find correct angle values.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions
Reference Triangles and Quadrants
Reference triangles help relate trigonometric values to angles in different quadrants. Knowing the sign of the trigonometric function in each quadrant allows you to determine the correct angle corresponding to a given value, especially when the input is negative.
Recommended video:
5:30Trig Values in Quadrants II, III, & IV
Secant Function and Its Properties
The secant function is the reciprocal of cosine, sec(θ) = 1/cos(θ). Recognizing this relationship helps in constructing reference triangles and understanding the behavior of arcsec, including its domain (|x| ≥ 1) and how to find angles from secant values.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
3. c. sin^(-1)(-√3/2)
Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
c. 1/√x
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?
c. ln(√x)
Textbook Question
In Exercises 1–4, solve for t.
1. c. e^((ln 0.2)t) = 0.4
Textbook Question
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
c. 3ln ∛(t² - 1) - ln(t+1)