Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
19. y = (sech θ)(1-ln(sech θ))
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.14
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
14. lim (t → 0) sin 5t / 2t
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Identify the limit expression: \(\lim_{t \to 0} \frac{\sin 5t}{2t}\).
Check if the limit is an indeterminate form by substituting \(t = 0\): both numerator \(\sin 5(0) = 0\) and denominator \(2(0) = 0\), so the form is \(\frac{0}{0}\), which allows the use of l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(t\): the derivative of the numerator \(\sin 5t\) is \(5 \cos 5t\), and the derivative of the denominator \$2t$ is \(2\).
Rewrite the limit using these derivatives: \(\lim_{t \to 0} \frac{5 \cos 5t}{2}\).
Evaluate the new limit by substituting \(t = 0\): calculate \(\frac{5 \cos 0}{2}\), noting that \(\cos 0 = 1\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits and Indeterminate Forms
Limits describe the behavior of a function as the input approaches a particular value. When direct substitution results in an indeterminate form like 0/0, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.
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l’Hôpital’s Rule
l’Hôpital’s rule states that if a limit yields an indeterminate form 0/0 or ∞/∞, the limit of the ratio of the functions equals the limit of the ratio of their derivatives, provided this latter limit exists.
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Derivative of Trigonometric Functions
Understanding the derivatives of sine and cosine functions is essential when applying l’Hôpital’s rule to trigonometric limits. For example, the derivative of sin(kt) with respect to t is k cos(kt), which helps simplify the limit expression.
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Related Practice
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