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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.2.4a
Capitolo 7, Problema 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

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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 \(\ln (1)\).

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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.
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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.
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Introduction 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.
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Logarithms Introduction
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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)

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4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

a. x² + √x

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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

69. ∫(from 5/4 to 2)dx/(1-x²)

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154. The linearization of log₃x

a. Find the linearization of

f(x) = log₃xatx = 3.

Then round its coefficients to two decimal places.

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In Exercises 41–44:

a. Find f⁻¹(x).


44. f(x) = 2x², x ≥ 0, a = 5

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