Textbook Question
Verify that the following functions satisfy the conditions of Theorem 4.9 on their domains. Then find the location and value of the absolute extrema guaranteed by the theorem.
f(x) = x√(3-x)
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.7.25
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ e (ln x - 1) / (x - 1)
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First, identify the form of the limit as x approaches e. Substitute x = e into the expression (ln x - 1) / (x - 1). You will find that both the numerator and the denominator approach 0, which is an indeterminate form 0/0.
Since the limit is in the indeterminate form 0/0, l'Hôpital's Rule can be applied. According to l'Hôpital's Rule, if the limit of f(x)/g(x) as x approaches a is in the form 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator.
Differentiate the numerator ln(x) - 1 with respect to x. The derivative of ln(x) is 1/x, and the derivative of the constant -1 is 0. Therefore, the derivative of the numerator is 1/x.
Differentiate the denominator x - 1 with respect to x. The derivative of x is 1, and the derivative of the constant -1 is 0. Therefore, the derivative of the denominator is 1.
Apply l'Hôpital's Rule by substituting the derivatives into the limit expression. The limit becomes lim_x→e (1/x) / 1. Simplify this expression to lim_x→e 1/x. Now, substitute x = e into the simplified expression to evaluate the limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this question, evaluating the limit as x approaches e involves determining the behavior of the function ln(x) - 1 over x - 1 near that point.
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One-Sided Limits
l'Hôpital's Rule
l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in the given limit problem when direct substitution leads to an indeterminate form.
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Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a crucial function in calculus, especially in limits and derivatives, due to its unique properties, such as ln(e) = 1 and its derivative being 1/x. In this limit problem, understanding the behavior of ln(x) as x approaches e is key to solving the limit.
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Related Practice
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (sec² Θ + sec Θ tan Θ)dΘ
Textbook Question
Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.
Textbook Question
Avalanche forecasting Avalanche forecasters measure the temperature gradient dT/dh, which is the rate at which the temperature in a snowpack T changes with respect to its depth h. A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters use the following rule of thumb: If dT/dh exceeds 10° C/m anywhere in the snowpack, conditions are favorable for weak-layer formation, and the risk of avalanche increases. Assume the temperature function is continuous and differentiable.
a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0), the temperature is -16° C. At a depth of 1.1 m, the temperature is -2° C. Using the Mean Value Theorem, what can he conclude about the temperature gradient? Is the formation of a weak layer likely?
Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 1/(e⁻ˣ - 1)
Textbook Question
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = eˣ + e⁻ˣ