Domanda del libro di testo
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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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.4.44
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 1/(e⁻ˣ - 1)
Guida verificata passo dopo passo1
Identify the domain of the function f(x) = \(\frac{1}{e^{-x} - 1}\). The expression e^{-x} - 1 cannot be zero, so solve e^{-x} - 1 = 0 to find the values of x that are not in the domain.
Determine the vertical asymptotes by finding the values of x where the denominator is zero. Since e^{-x} = 1 when x = 0, there is a vertical asymptote at x = 0.
Analyze the behavior of the function as x approaches the vertical asymptote from both sides. As x approaches 0 from the left, e^{-x} approaches infinity, making the denominator approach infinity, and thus f(x) approaches 0. As x approaches 0 from the right, e^{-x} approaches 1, making the denominator approach 0, and thus f(x) approaches infinity.
Examine the end behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, e^{-x} approaches 0, making the denominator approach -1, so f(x) approaches -1. As x approaches negative infinity, e^{-x} approaches infinity, making the denominator approach infinity, so f(x) approaches 0.
Identify any horizontal asymptotes based on the end behavior. From the analysis, as x approaches positive infinity, f(x) approaches -1, indicating a horizontal asymptote at y = -1. As x approaches negative infinity, f(x) approaches 0, indicating a horizontal asymptote at y = 0.
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Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f(x) = 1/(e⁻ˣ - 1), the domain excludes values where the denominator is zero, i.e., e⁻ˣ - 1 = 0, which occurs at x = 0. Thus, the domain is all real numbers except x = 0.
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Percorso guidato
5:10
Finding the Domain and Range of a Graph
Asymptotes
Asymptotes are lines that a graph approaches but never touches. For the function f(x) = 1/(e⁻ˣ - 1), there is a vertical asymptote at x = 0, where the function is undefined. Additionally, as x approaches positive or negative infinity, the function approaches horizontal asymptotes, which can be determined by analyzing the behavior of the function as x becomes very large or very small.
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5:37Introduction to Cotangent Graph
Behavior at Infinity
Understanding the behavior of a function as x approaches infinity or negative infinity helps in sketching its graph. For f(x) = 1/(e⁻ˣ - 1), as x approaches positive infinity, e⁻ˣ approaches zero, making the function approach -1. As x approaches negative infinity, e⁻ˣ becomes very large, making the function approach zero. This behavior helps in identifying horizontal asymptotes and the overall shape of the graph.
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03:07Cases Where Limits Do Not Exist
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Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.
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Domanda del libro di testo
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?
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Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
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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