Textbook Question
Solve the initial value problems in Exercises 55–58.
55. dy/dt = e^t sin(e^t − 2),y(ln 2) = 0
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 7.5.9
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
9. lim (t → -3) (t³ - 4t + 15) / (t² - t - 12)
Verified step by step guidance1
First, identify the limit expression: \(\lim_{t \to -3} \frac{t^{3} - 4t + 15}{t^{2} - t - 12}\).
Evaluate the numerator and denominator separately at \(t = -3\) to check if the limit is an indeterminate form. Calculate \((-3)^{3} - 4(-3) + 15\) and \((-3)^{2} - (-3) - 12\).
If both numerator and denominator evaluate to 0, then the limit is of the form \(\frac{0}{0}\), and l’Hôpital’s Rule can be applied.
Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(t\): find \(\frac{d}{dt}(t^{3} - 4t + 15)\) and \(\frac{d}{dt}(t^{2} - t - 12)\).
After differentiation, substitute \(t = -3\) into the new expression \(\frac{\frac{d}{dt}(\text{numerator})}{\frac{d}{dt}(\text{denominator})}\) to find the limit.
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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 or ∞/∞, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.
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One-Sided Limits
l’Hôpital’s Rule
l’Hôpital’s rule provides a method to evaluate limits that yield indeterminate forms by differentiating the numerator and denominator separately and then taking the limit of their quotient. It applies only when the original limit results in 0/0 or ∞/∞.
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Differentiation of Polynomial Functions
Differentiation involves finding the derivative of a function, which represents its rate of change. For polynomials, derivatives are found by applying the power rule term-by-term, which is essential when using l’Hôpital’s rule to differentiate numerator and denominator.
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