Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = xe⁻(ˣ²/₂)
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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 4.7.27
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (1 - ln x) / (1 + ln x)
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First, identify the form of the limit as x approaches 0 from the positive side. Substitute x = 0⁺ into the expression (1 - ln x) / (1 + ln x) to check if it results in an indeterminate form like 0/0 or ∞/∞.
Since ln(x) approaches -∞ as x approaches 0⁺, both the numerator (1 - ln x) and the denominator (1 + ln x) approach ∞, resulting in the indeterminate form ∞/∞. This indicates that l'Hôpital's Rule can be applied.
Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, then the limit can be evaluated as the limit of f'(x)/g'(x) as x approaches the same point, provided the derivatives exist.
Differentiate the numerator: The derivative of 1 - ln x with respect to x is -1/x.
Differentiate the denominator: The derivative of 1 + ln x with respect to x is 1/x.
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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 concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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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 separately. This rule simplifies the process of finding limits in complex functions.
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5:50Power Rules
Natural Logarithm (ln)
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in limits and derivatives, due to its unique properties, such as ln(1) = 0 and its derivative being 1/x. Understanding the behavior of ln(x) as x approaches 0 is essential for evaluating limits involving this function.
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Related Practice
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
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x⁴/4 - 8x³/3 + 15x²/2 + 8
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Textbook Question
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>
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Textbook Question
Solving initial value problems Find the solution of the following initial value problems.
y'(Θ) = ((√2 cos³ Θ + 1)/cos² Θ); y (π/4) = 3, -π/2 < Θ < π/2