Textbook Question
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
a. f(x) = x, g(x) = x², (a, b) = (-2, 0)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 7.2.4a
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
a. ln secθ + ln cosθ
Verified step by step guidance1
Recall the logarithm property that states: \(\ln a + \ln b = \ln (a \times b)\). This means the sum of two logarithms with the same base can be combined into the logarithm of the product of their arguments.
Identify the expressions inside the logarithms: here, we have \(\ln \sec \theta\) and \(\ln \cos \theta\).
Apply the property by multiplying the arguments inside the logarithms: \(\ln (\sec \theta \times \cos \theta)\).
Simplify the product inside the logarithm. Recall that \(\sec \theta = \frac{1}{\cos \theta}\), so multiply \(\sec \theta\) and \(\cos \theta\) accordingly.
Write the final expression as a single logarithm term: \(\ln (\sec \theta \cdot \cos \theta)\), which simplifies to \(\ln (1)\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Logarithmic properties include rules such as the product, quotient, and power rules. The product rule states that the sum of logarithms with the same base can be combined into the logarithm of the product of their arguments, i.e., ln(a) + ln(b) = ln(ab). This is essential for rewriting expressions as a single logarithmic term.
Recommended video:
05:36
Change of Base Property
Trigonometric Functions and Identities
Understanding basic trigonometric functions like secant (sec θ = 1/cos θ) and cosine is crucial. Recognizing these relationships allows simplification of expressions inside logarithms, which helps in combining or reducing terms effectively.
Recommended video:
6:04Introduction to Trigonometric Functions
Simplification of Logarithmic Expressions
Simplifying logarithmic expressions involves applying logarithm properties and algebraic manipulation to rewrite complex expressions into simpler or single terms. This skill is important for solving problems efficiently and for clearer interpretation of results.
Recommended video:
7:30Logarithms Introduction
Related Practice
Textbook Question
143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.
Textbook Question
1. 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. x-3
1
views
Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
69. ∫(from 5/4 to 2)dx/(1-x²)
Textbook Question
10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)
Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
44. f(x) = 2x², x ≥ 0, a = 5