Textbook Question
{Use of Tech} Finding all roots Use Newtonβs method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = xΒ²(x - 100) + 1
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4. Applications of Derivatives
Differentials
4:47 minutesProblem 28
Textbook Question
Calculate the first derivatives of Ζ(π) = πΒ²/ (πΒ² + 1) and g(π) = β1/ (πΒ² + 1) . What can you conclude about the graphs of these functions?
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To find the first derivative of Ζ(π) = πΒ² / (πΒ² + 1), apply the quotient rule. The quotient rule states that if you have a function h(π) = u(π) / v(π), then its derivative h'(π) is given by (u'(π)v(π) - u(π)v'(π)) / (v(π)Β²). Here, u(π) = πΒ² and v(π) = πΒ² + 1.
Calculate the derivatives of u(π) and v(π). The derivative of u(π) = πΒ² is u'(π) = 2π. The derivative of v(π) = πΒ² + 1 is v'(π) = 2π.
Substitute these derivatives into the quotient rule formula: Ζ'(π) = ((2π)(πΒ² + 1) - (πΒ²)(2π)) / (πΒ² + 1)Β².
For g(π) = -1 / (πΒ² + 1), use the chain rule. The chain rule states that if you have a composite function y = f(g(π)), then its derivative is y' = f'(g(π)) * g'(π). Here, consider g(π) as -1 times the reciprocal of (πΒ² + 1).
The derivative of g(π) = -1 / (πΒ² + 1) is g'(π) = (0 * (πΒ² + 1) - (-1)(2π)) / (πΒ² + 1)Β² = 2π / (πΒ² + 1)Β². Compare the derivatives of Ζ(π) and g(π) to analyze the behavior of their graphs, noting that they are related by symmetry or transformations.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the graph of the function at any given point. Derivatives can be calculated using rules such as the quotient rule, which is particularly useful for functions defined as a ratio of two other functions.
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Derivatives
Quotient Rule
The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions. If you have a function defined as Ζ(π) = u(π)/v(π), the derivative is given by Ζ'(π) = (u'v - uv')/vΒ², where u' and v' are the derivatives of u and v, respectively. This rule is essential for differentiating functions like Ζ(π) = πΒ²/(πΒ² + 1) and g(π) = -1/(πΒ² + 1).
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Graph Behavior and Critical Points
Analyzing the first derivative of a function helps determine its critical points, where the function's slope is zero or undefined. These points are crucial for understanding the behavior of the graph, such as identifying local maxima, minima, and points of inflection. By evaluating the first derivatives of Ζ(π) and g(π), one can draw conclusions about the increasing or decreasing nature of the functions and their overall shape.
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