Domanda del libro di testo
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.3.131b
131. Let f(x) = x * e^(−x).
b. Find all inflection points for f.
Guida verificata passo dopo passo1
Recall that inflection points occur where the concavity of the function changes, which means where the second derivative changes sign. So, first, we need to find the second derivative of the function \(f(x) = x \cdot e^{-x}\).
Start by finding the first derivative \(f'(x)\). Use the product rule: if \(f(x) = u(x) \cdot v(x)\), then \(f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)\). Here, \(u(x) = x\) and \(v(x) = e^{-x}\).
Calculate \(u'(x) = 1\) and \(v'(x) = -e^{-x}\) (using the chain rule). Substitute these into the product rule formula to get \(f'(x) = 1 \cdot e^{-x} + x \cdot (-e^{-x})\).
Simplify \(f'(x)\) to \(f'(x) = e^{-x} - x e^{-x} = e^{-x}(1 - x)\). Next, find the second derivative \(f''(x)\) by differentiating \(f'(x)\) again, applying the product rule to \(e^{-x}(1 - x)\).
Set \(f''(x) = 0\) and solve for \(x\) to find potential inflection points. Then, check the sign of \(f''(x)\) on intervals around these points to confirm where the concavity changes, identifying the inflection points.
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Second Derivative and Concavity
The second derivative of a function measures the curvature or concavity of its graph. If the second derivative is positive on an interval, the graph is concave up; if negative, concave down. Inflection points occur where the concavity changes, typically where the second derivative equals zero or is undefined.
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06:02
The Second Derivative Test: Finding Local Extrema
Finding Inflection Points
To find inflection points, first compute the second derivative of the function, then solve for points where it equals zero or does not exist. Finally, verify that the concavity changes sign around these points to confirm they are true inflection points.
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04:50Critical Points
Product Rule for Differentiation
Since f(x) = x * e^(−x) is a product of two functions, the product rule is used to find its derivatives. The product rule states that the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x), which is essential for correctly computing the first and second derivatives.
Video consigliato:
05:18The Product Rule
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