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Questions and Answers
If $f'(x) = -g(x)$ and $g(x) = x + 2\ln x$, what can be said about the relationship between $f(x)$ and $x + 2\ln x$?
If $f'(x) = -g(x)$ and $g(x) = x + 2\ln x$, what can be said about the relationship between $f(x)$ and $x + 2\ln x$?
- $f(x)$ is the negative derivative of $x + 2\ln x$
- $f(x)$ is the antiderivative of $x + 2\ln x$
- $f(x)$ is the derivative of $x + 2\ln x$
- $f(x)$ is the negative antiderivative of $x + 2\ln x$ (correct)
Given the function $g(x) = x + 2\ln x$, which of the following transformations would result in a vertical stretch by a factor of 3?
Given the function $g(x) = x + 2\ln x$, which of the following transformations would result in a vertical stretch by a factor of 3?
- $g(x) = x + 6\ln x$
- $3g(x) = 3x + 6\ln x$ (correct)
- $g(x) = 3x + 2\ln x$
- $g(3x) = 3x + 2\ln (3x)$
Which of the following is the correct interpretation of the notation $f'(x)$?
Which of the following is the correct interpretation of the notation $f'(x)$?
- The integral of the function $f(x)$
- The derivative of the function $f(x)$ (correct)
- The reciprocal of the function $f(x)$
- The inverse of the function $f(x)$
Consider the function $f(x) = 1 - x + \frac{1}{2}(1 + \ln x)$. Which term primarily dictates the behavior of the function as $x$ approaches infinity?
Consider the function $f(x) = 1 - x + \frac{1}{2}(1 + \ln x)$. Which term primarily dictates the behavior of the function as $x$ approaches infinity?
Based on the equations $g(x) = x + 2\ln x$ and $f(x) = 1 - x + \frac{1}{2}(1 + \ln x)$, which of the following statements is necessarily true?
Based on the equations $g(x) = x + 2\ln x$ and $f(x) = 1 - x + \frac{1}{2}(1 + \ln x)$, which of the following statements is necessarily true?
Flashcards
Mathematical Function
Mathematical Function
A relation between a set of inputs and outputs where each input has exactly one output.
Natural Logarithm
Natural Logarithm
The logarithm to the base 'e', where 'e' is approximately 2.71828, commonly used in calculus.
Derivative
Derivative
A measure of how a function changes as its input changes, the slope of the tangent line.
Equation $g(x) = x + 2 ext{ln }x$
Equation $g(x) = x + 2 ext{ln }x$
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Equation $f'(x) = -g(x)$
Equation $f'(x) = -g(x)$
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Study Notes
Function Analysis
- A function g(x) is defined as g(x) = x + 2lnx
- The domain of g(x) is 0.75 < x < 0.176
- The function f(x) is defined as f(x) = 1 - x + (1/2)(1 + ln x)
- The derivative of f(x) is f'(x) = -g(x)/x
- f(3) is calculated as 1 - 3 + (1/2)(1 + ln3)
- f(1) is calculated as 1 - 1 + (1/2)(1 + ln1), and ln 1 equals 0; therefore, f(1) = 1/2
- x = 3 is a value
- x = 1 is a value
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